diff --git a/src/interval/interval.h b/src/interval/interval.h
index 28f51a138..163a3c74d 100644
--- a/src/interval/interval.h
+++ b/src/interval/interval.h
@@ -168,8 +168,8 @@ public:
     void sin  ();
     void cos  ();
     void tan  ();
-    void sec  ();
     void csc  ();
+    void sec  ();
     void cot  ();
     void asin ();
     void acos ();
@@ -191,8 +191,8 @@ public:
     friend interval<T> sin  (interval<T> o) { o.sin();   return o; }
     friend interval<T> cos  (interval<T> o) { o.cos();   return o; }
     friend interval<T> tan  (interval<T> o) { o.tan();   return o; }
-    friend interval<T> sec  (interval<T> o) { o.sec();   return o; }
     friend interval<T> csc  (interval<T> o) { o.csc();   return o; }
+    friend interval<T> sec  (interval<T> o) { o.sec();   return o; }
     friend interval<T> cot  (interval<T> o) { o.cot();   return o; }
     friend interval<T> asin (interval<T> o) { o.asin();  return o; }
     friend interval<T> acos (interval<T> o) { o.acos();  return o; }
diff --git a/src/interval/interval_def.h b/src/interval/interval_def.h
index 9842d2418..91d95b5bf 100644
--- a/src/interval/interval_def.h
+++ b/src/interval/interval_def.h
@@ -1250,9 +1250,147 @@ template<typename T> void interval<T>::tan() {
     }
 }
 
-template<typename T> void interval<T>::sec  () { /* TODO */ lean_unreachable(); return; }
-template<typename T> void interval<T>::csc  () { /* TODO */ lean_unreachable(); return; }
-template<typename T> void interval<T>::cot  () { /* TODO */ lean_unreachable(); return; }
+template<typename T> void interval<T>::csc  () {
+    // csc(x) = 1 / sin(x)
+    if(m_lower_inf || m_upper_inf || (m_upper - m_lower > numeric_traits<T>::pi())) {
+        // csc([-oo, c]) = [-oo, +oo]
+        // csc([c, +oo]) = [-oo, +oo]
+        // if the width is bigger than pi, then the result is [-oo, +oo]
+        numeric_traits<T>::reset(m_lower);
+        numeric_traits<T>::reset(m_upper);
+        m_lower_open = m_upper_open = true;
+        m_lower_inf  = m_upper_inf = true;
+        lean_assert(check_invariant());
+        return;
+    }
+    fmod(interval<T>(numeric_traits<T>::pi_twice_lower(), numeric_traits<T>::pi_twice_upper()));
+    if(m_upper > numeric_traits<T>::pi_twice() ||
+       (m_lower < numeric_traits<T>::pi() && numeric_traits<T>::pi() < m_upper)) {
+        // l < 2pi < u or l < pi < u
+        // then the result = [-oo, +oo]
+        numeric_traits<T>::reset(m_lower);
+        numeric_traits<T>::reset(m_upper);
+        m_lower_open = m_upper_open = true;
+        m_lower_inf  = m_upper_inf = true;
+        lean_assert(check_invariant());
+        return;
+    }
+
+    if (m_lower <= numeric_traits<T>::pi_half()) {
+        if (m_upper <= numeric_traits<T>::pi_half()) {
+            // l <= u <= 1/2 pi
+            // csc[l, u] = [csc(u), csc(l)]
+            std::swap(m_lower, m_upper);
+            numeric_traits<T>::set_rounding(false);
+            numeric_traits<T>::csc(m_lower);
+            numeric_traits<T>::set_rounding(true);
+            numeric_traits<T>::csc(m_upper);
+            lean_assert(check_invariant());
+            return;
+        }
+        if (m_upper <= numeric_traits<T>::pi()) {
+            // l <= 1/2 pi <= u <= pi
+            // csc[l, u] = [1, max(csc(l), csc(u))]
+            //           = [1, csc(l)]     if l + u <= pi
+            //           = [1, csc(u)]     if l + u >= pi
+            if (m_lower + m_upper < numeric_traits<T>::pi()) {
+                m_upper = m_lower;
+            } else {
+                // Nothing
+            }
+            m_lower = 1.0;
+            numeric_traits<T>::set_rounding(true);
+            numeric_traits<T>::csc(m_upper);
+            lean_assert(check_invariant());
+            return;
+        }
+        lean_unreachable();
+        return;
+    }
+
+    if (m_lower <= numeric_traits<T>::pi() && m_upper <= numeric_traits<T>::pi()) {
+        // l <= u <= pi
+        // csc[l, u] = [csc(l), csc(u)]
+        numeric_traits<T>::set_rounding(false);
+        numeric_traits<T>::csc(m_lower);
+        numeric_traits<T>::set_rounding(true);
+        numeric_traits<T>::csc(m_upper);
+        lean_assert(check_invariant());
+        return;
+    }
+    if (m_lower <= numeric_traits<T>::pi() + numeric_traits<T>::pi_half()) {
+        if (m_upper <= numeric_traits<T>::pi() + numeric_traits<T>::pi_half()) {
+            // l <= u <= 3/2 pi
+            // csc[l, u] = [csc(l), csc(u)]
+            numeric_traits<T>::set_rounding(false);
+            numeric_traits<T>::csc(m_lower);
+            numeric_traits<T>::set_rounding(true);
+            numeric_traits<T>::csc(m_upper);
+            lean_assert(check_invariant());
+            return;
+        }
+        // l <= 3/2 pi <= u <= 2pi
+        // csc[l, u] = [min(csc(l), csc(u)), -1]
+        //           = [csc(l), -1]     if l + u <= 3pi
+        //           = [csc(u), -1]     if l + u >= 3pi
+        if (m_lower + m_upper < numeric_traits<T>::pi() + numeric_traits<T>::pi_twice()) {
+            // Nothing
+        } else {
+            m_lower = m_upper;
+        }
+        m_upper = -1.0;
+        numeric_traits<T>::set_rounding(false);
+        numeric_traits<T>::csc(m_lower);
+        lean_assert(check_invariant());
+        return;
+    }
+    // 3/2pi <= l <= u < 2pi
+    lean_assert(numeric_traits<T>::pi_half() + numeric_traits<T>::pi() < m_lower);
+    lean_assert(m_upper < numeric_traits<T>::pi_twice());
+    std::swap(m_lower, m_upper);
+    numeric_traits<T>::set_rounding(false);
+    numeric_traits<T>::csc(m_lower);
+    numeric_traits<T>::set_rounding(true);
+    numeric_traits<T>::csc(m_upper);
+    lean_assert(check_invariant());
+    return;
+}
+
+template<typename T> void interval<T>::sec  () {
+    *this += numeric_traits<T>::pi_half();
+    csc();
+    return;
+}
+
+template<typename T> void interval<T>::cot  () {
+    if(m_lower_inf || m_upper_inf) {
+        // cot([-oo, c]) = [-oo, +oo]
+        // cot([c, +oo]) = [-oo, +oo]
+        numeric_traits<T>::reset(m_lower);
+        numeric_traits<T>::reset(m_upper);
+        m_lower_open = m_upper_open = true;
+        m_lower_inf  = m_upper_inf = true;
+        lean_assert(check_invariant());
+        return;
+    }
+    fmod(interval<T>(numeric_traits<T>::pi_lower(), numeric_traits<T>::pi_upper()));
+    if (m_upper >= numeric_traits<T>::pi()) {
+        numeric_traits<T>::reset(m_lower);
+        numeric_traits<T>::reset(m_upper);
+        m_lower_open = m_upper_open = true;
+        m_lower_inf  = m_upper_inf = true;
+        lean_assert(check_invariant());
+        return;
+    }
+    std::swap(m_lower, m_upper);
+    numeric_traits<T>::set_rounding(false);
+    numeric_traits<T>::cot(m_lower);
+    numeric_traits<T>::set_rounding(true);
+    numeric_traits<T>::cot(m_upper);
+    lean_assert(check_invariant());
+    return;
+}
+
 template<typename T> void interval<T>::asin () {
     lean_assert(lower_kind() == XN_NUMERAL && upper_kind() == XN_NUMERAL);
     lean_assert(-1.0 <= m_lower && m_upper <= 1.0);
diff --git a/src/tests/interval/double_interval.cpp b/src/tests/interval/double_interval.cpp
index 6ad14eee8..abbd72125 100644
--- a/src/tests/interval/double_interval.cpp
+++ b/src/tests/interval/double_interval.cpp
@@ -855,6 +855,117 @@ static void double_interval_trans() {
     check_uop(double, tanh, ozero_ninf);
     check_uop(double, tanh, czero_pinf);
     check_uop(double, tanh, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(double, csc, i1);
+    check_uop(double, csc, i2);
+    check_uop(double, csc, i3);
+    check_uop(double, csc, i4);
+    check_uop(double, csc, i5);
+    check_uop(double, csc, i6);
+    check_uop(double, csc, i7);
+    check_uop(double, csc, i8);
+    check_uop(double, csc, i9);
+    check_uop(double, csc, i10);
+    check_uop(double, csc, i11);
+    check_uop(double, csc, i12);
+    check_uop(double, csc, i13);
+    check_uop(double, csc, i14);
+    check_uop(double, csc, i15);
+    check_uop(double, csc, oi1);
+    check_uop(double, csc, oi2);
+    check_uop(double, csc, oi3);
+    check_uop(double, csc, oi4);
+    check_uop(double, csc, oi5);
+    check_uop(double, csc, oi6);
+    check_uop(double, csc, oi7);
+    check_uop(double, csc, oi8);
+    check_uop(double, csc, oi9);
+    check_uop(double, csc, oi10);
+    check_uop(double, csc, oi11);
+    check_uop(double, csc, oi12);
+    check_uop(double, csc, oi13);
+    check_uop(double, csc, oi14);
+    check_uop(double, csc, oi15);
+    check_uop(double, csc, inf);
+    check_uop(double, csc, ozero_pinf);
+    check_uop(double, csc, ozero_ninf);
+    check_uop(double, csc, czero_pinf);
+    check_uop(double, csc, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(double, sec, i1);
+    check_uop(double, sec, i2);
+    check_uop(double, sec, i3);
+    check_uop(double, sec, i4);
+    check_uop(double, sec, i5);
+    check_uop(double, sec, i6);
+    check_uop(double, sec, i7);
+    check_uop(double, sec, i8);
+    check_uop(double, sec, i9);
+    check_uop(double, sec, i10);
+    check_uop(double, sec, i11);
+    check_uop(double, sec, i12);
+    check_uop(double, sec, i13);
+    check_uop(double, sec, i14);
+    check_uop(double, sec, i15);
+    check_uop(double, sec, oi1);
+    check_uop(double, sec, oi2);
+    check_uop(double, sec, oi3);
+    check_uop(double, sec, oi4);
+    check_uop(double, sec, oi5);
+    check_uop(double, sec, oi6);
+    check_uop(double, sec, oi7);
+    check_uop(double, sec, oi8);
+    check_uop(double, sec, oi9);
+    check_uop(double, sec, oi10);
+    check_uop(double, sec, oi11);
+    check_uop(double, sec, oi12);
+    check_uop(double, sec, oi13);
+    check_uop(double, sec, oi14);
+    check_uop(double, sec, oi15);
+    check_uop(double, sec, inf);
+    check_uop(double, sec, ozero_pinf);
+    check_uop(double, sec, ozero_ninf);
+    check_uop(double, sec, czero_pinf);
+    check_uop(double, sec, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(double, cot, i1);
+    check_uop(double, cot, i2);
+    check_uop(double, cot, i3);
+    check_uop(double, cot, i4);
+    check_uop(double, cot, i5);
+    check_uop(double, cot, i6);
+    check_uop(double, cot, i7);
+    check_uop(double, cot, i8);
+    check_uop(double, cot, i9);
+    check_uop(double, cot, i10);
+    check_uop(double, cot, i11);
+    check_uop(double, cot, i12);
+    check_uop(double, cot, i13);
+    check_uop(double, cot, i14);
+    check_uop(double, cot, i15);
+    check_uop(double, cot, oi1);
+    check_uop(double, cot, oi2);
+    check_uop(double, cot, oi3);
+    check_uop(double, cot, oi4);
+    check_uop(double, cot, oi5);
+    check_uop(double, cot, oi6);
+    check_uop(double, cot, oi7);
+    check_uop(double, cot, oi8);
+    check_uop(double, cot, oi9);
+    check_uop(double, cot, oi10);
+    check_uop(double, cot, oi11);
+    check_uop(double, cot, oi12);
+    check_uop(double, cot, oi13);
+    check_uop(double, cot, oi14);
+    check_uop(double, cot, oi15);
+    check_uop(double, cot, inf);
+    check_uop(double, cot, ozero_pinf);
+    check_uop(double, cot, ozero_ninf);
+    check_uop(double, cot, czero_pinf);
+    check_uop(double, cot, czero_ninf);
 }
 
 int main() {
diff --git a/src/tests/interval/float_interval.cpp b/src/tests/interval/float_interval.cpp
index 211836a9e..8bb0f3746 100644
--- a/src/tests/interval/float_interval.cpp
+++ b/src/tests/interval/float_interval.cpp
@@ -855,6 +855,117 @@ static void float_interval_trans() {
     check_uop(float, tanh, ozero_ninf);
     check_uop(float, tanh, czero_pinf);
     check_uop(float, tanh, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(float, csc, i1);
+    check_uop(float, csc, i2);
+    check_uop(float, csc, i3);
+    check_uop(float, csc, i4);
+    check_uop(float, csc, i5);
+    check_uop(float, csc, i6);
+    check_uop(float, csc, i7);
+    check_uop(float, csc, i8);
+    check_uop(float, csc, i9);
+    check_uop(float, csc, i10);
+    check_uop(float, csc, i11);
+    check_uop(float, csc, i12);
+    check_uop(float, csc, i13);
+    check_uop(float, csc, i14);
+    check_uop(float, csc, i15);
+    check_uop(float, csc, oi1);
+    check_uop(float, csc, oi2);
+    check_uop(float, csc, oi3);
+    check_uop(float, csc, oi4);
+    check_uop(float, csc, oi5);
+    check_uop(float, csc, oi6);
+    check_uop(float, csc, oi7);
+    check_uop(float, csc, oi8);
+    check_uop(float, csc, oi9);
+    check_uop(float, csc, oi10);
+    check_uop(float, csc, oi11);
+    check_uop(float, csc, oi12);
+    check_uop(float, csc, oi13);
+    check_uop(float, csc, oi14);
+    check_uop(float, csc, oi15);
+    check_uop(float, csc, inf);
+    check_uop(float, csc, ozero_pinf);
+    check_uop(float, csc, ozero_ninf);
+    check_uop(float, csc, czero_pinf);
+    check_uop(float, csc, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(float, sec, i1);
+    check_uop(float, sec, i2);
+    check_uop(float, sec, i3);
+    check_uop(float, sec, i4);
+    check_uop(float, sec, i5);
+    check_uop(float, sec, i6);
+    check_uop(float, sec, i7);
+    check_uop(float, sec, i8);
+    check_uop(float, sec, i9);
+    check_uop(float, sec, i10);
+    check_uop(float, sec, i11);
+    check_uop(float, sec, i12);
+    check_uop(float, sec, i13);
+    check_uop(float, sec, i14);
+    check_uop(float, sec, i15);
+    check_uop(float, sec, oi1);
+    check_uop(float, sec, oi2);
+    check_uop(float, sec, oi3);
+    check_uop(float, sec, oi4);
+    check_uop(float, sec, oi5);
+    check_uop(float, sec, oi6);
+    check_uop(float, sec, oi7);
+    check_uop(float, sec, oi8);
+    check_uop(float, sec, oi9);
+    check_uop(float, sec, oi10);
+    check_uop(float, sec, oi11);
+    check_uop(float, sec, oi12);
+    check_uop(float, sec, oi13);
+    check_uop(float, sec, oi14);
+    check_uop(float, sec, oi15);
+    check_uop(float, sec, inf);
+    check_uop(float, sec, ozero_pinf);
+    check_uop(float, sec, ozero_ninf);
+    check_uop(float, sec, czero_pinf);
+    check_uop(float, sec, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(float, cot, i1);
+    check_uop(float, cot, i2);
+    check_uop(float, cot, i3);
+    check_uop(float, cot, i4);
+    check_uop(float, cot, i5);
+    check_uop(float, cot, i6);
+    check_uop(float, cot, i7);
+    check_uop(float, cot, i8);
+    check_uop(float, cot, i9);
+    check_uop(float, cot, i10);
+    check_uop(float, cot, i11);
+    check_uop(float, cot, i12);
+    check_uop(float, cot, i13);
+    check_uop(float, cot, i14);
+    check_uop(float, cot, i15);
+    check_uop(float, cot, oi1);
+    check_uop(float, cot, oi2);
+    check_uop(float, cot, oi3);
+    check_uop(float, cot, oi4);
+    check_uop(float, cot, oi5);
+    check_uop(float, cot, oi6);
+    check_uop(float, cot, oi7);
+    check_uop(float, cot, oi8);
+    check_uop(float, cot, oi9);
+    check_uop(float, cot, oi10);
+    check_uop(float, cot, oi11);
+    check_uop(float, cot, oi12);
+    check_uop(float, cot, oi13);
+    check_uop(float, cot, oi14);
+    check_uop(float, cot, oi15);
+    check_uop(float, cot, inf);
+    check_uop(float, cot, ozero_pinf);
+    check_uop(float, cot, ozero_ninf);
+    check_uop(float, cot, czero_pinf);
+    check_uop(float, cot, czero_ninf);
 }
 
 int main() {
diff --git a/src/tests/interval/mpfp_interval.cpp b/src/tests/interval/mpfp_interval.cpp
index b48188a57..27bf2c4b9 100644
--- a/src/tests/interval/mpfp_interval.cpp
+++ b/src/tests/interval/mpfp_interval.cpp
@@ -855,6 +855,117 @@ static void mpfp_interval_trans() {
     check_uop(mpfp, tanh, ozero_ninf);
     check_uop(mpfp, tanh, czero_pinf);
     check_uop(mpfp, tanh, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(mpfp, csc, i1);
+    check_uop(mpfp, csc, i2);
+    check_uop(mpfp, csc, i3);
+    check_uop(mpfp, csc, i4);
+    check_uop(mpfp, csc, i5);
+    check_uop(mpfp, csc, i6);
+    check_uop(mpfp, csc, i7);
+    check_uop(mpfp, csc, i8);
+    check_uop(mpfp, csc, i9);
+    check_uop(mpfp, csc, i10);
+    check_uop(mpfp, csc, i11);
+    check_uop(mpfp, csc, i12);
+    check_uop(mpfp, csc, i13);
+    check_uop(mpfp, csc, i14);
+    check_uop(mpfp, csc, i15);
+    check_uop(mpfp, csc, oi1);
+    check_uop(mpfp, csc, oi2);
+    check_uop(mpfp, csc, oi3);
+    check_uop(mpfp, csc, oi4);
+    check_uop(mpfp, csc, oi5);
+    check_uop(mpfp, csc, oi6);
+    check_uop(mpfp, csc, oi7);
+    check_uop(mpfp, csc, oi8);
+    check_uop(mpfp, csc, oi9);
+    check_uop(mpfp, csc, oi10);
+    check_uop(mpfp, csc, oi11);
+    check_uop(mpfp, csc, oi12);
+    check_uop(mpfp, csc, oi13);
+    check_uop(mpfp, csc, oi14);
+    check_uop(mpfp, csc, oi15);
+    check_uop(mpfp, csc, inf);
+    check_uop(mpfp, csc, ozero_pinf);
+    check_uop(mpfp, csc, ozero_ninf);
+    check_uop(mpfp, csc, czero_pinf);
+    check_uop(mpfp, csc, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(mpfp, sec, i1);
+    check_uop(mpfp, sec, i2);
+    check_uop(mpfp, sec, i3);
+    check_uop(mpfp, sec, i4);
+    check_uop(mpfp, sec, i5);
+    check_uop(mpfp, sec, i6);
+    check_uop(mpfp, sec, i7);
+    check_uop(mpfp, sec, i8);
+    check_uop(mpfp, sec, i9);
+    check_uop(mpfp, sec, i10);
+    check_uop(mpfp, sec, i11);
+    check_uop(mpfp, sec, i12);
+    check_uop(mpfp, sec, i13);
+    check_uop(mpfp, sec, i14);
+    check_uop(mpfp, sec, i15);
+    check_uop(mpfp, sec, oi1);
+    check_uop(mpfp, sec, oi2);
+    check_uop(mpfp, sec, oi3);
+    check_uop(mpfp, sec, oi4);
+    check_uop(mpfp, sec, oi5);
+    check_uop(mpfp, sec, oi6);
+    check_uop(mpfp, sec, oi7);
+    check_uop(mpfp, sec, oi8);
+    check_uop(mpfp, sec, oi9);
+    check_uop(mpfp, sec, oi10);
+    check_uop(mpfp, sec, oi11);
+    check_uop(mpfp, sec, oi12);
+    check_uop(mpfp, sec, oi13);
+    check_uop(mpfp, sec, oi14);
+    check_uop(mpfp, sec, oi15);
+    check_uop(mpfp, sec, inf);
+    check_uop(mpfp, sec, ozero_pinf);
+    check_uop(mpfp, sec, ozero_ninf);
+    check_uop(mpfp, sec, czero_pinf);
+    check_uop(mpfp, sec, czero_ninf);
+
+    cout << "=====================" << endl;
+    check_uop(mpfp, cot, i1);
+    check_uop(mpfp, cot, i2);
+    check_uop(mpfp, cot, i3);
+    check_uop(mpfp, cot, i4);
+    check_uop(mpfp, cot, i5);
+    check_uop(mpfp, cot, i6);
+    check_uop(mpfp, cot, i7);
+    check_uop(mpfp, cot, i8);
+    check_uop(mpfp, cot, i9);
+    check_uop(mpfp, cot, i10);
+    check_uop(mpfp, cot, i11);
+    check_uop(mpfp, cot, i12);
+    check_uop(mpfp, cot, i13);
+    check_uop(mpfp, cot, i14);
+    check_uop(mpfp, cot, i15);
+    check_uop(mpfp, cot, oi1);
+    check_uop(mpfp, cot, oi2);
+    check_uop(mpfp, cot, oi3);
+    check_uop(mpfp, cot, oi4);
+    check_uop(mpfp, cot, oi5);
+    check_uop(mpfp, cot, oi6);
+    check_uop(mpfp, cot, oi7);
+    check_uop(mpfp, cot, oi8);
+    check_uop(mpfp, cot, oi9);
+    check_uop(mpfp, cot, oi10);
+    check_uop(mpfp, cot, oi11);
+    check_uop(mpfp, cot, oi12);
+    check_uop(mpfp, cot, oi13);
+    check_uop(mpfp, cot, oi14);
+    check_uop(mpfp, cot, oi15);
+    check_uop(mpfp, cot, inf);
+    check_uop(mpfp, cot, ozero_pinf);
+    check_uop(mpfp, cot, ozero_ninf);
+    check_uop(mpfp, cot, czero_pinf);
+    check_uop(mpfp, cot, czero_ninf);
 }
 
 int main() {