#### Question

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

#### Solution

It is known that if *g *and *h *are two continuous functions, then

It has to be proved first that *g* (*x*) = sin *x *and *h* (*x*) = cos *x* are continuous functions.

Let *g *(*x*) = sin *x*

It is evident that *g* (*x*) = sin *x* is defined for every real number.

Let *c *be a real number. Put *x* = *c* + *h*

If *x* → *c*, then *h* → 0

Therefore, *g* is a continuous function.

Let *h* (*x*) = cos *x*

It is evident that *h* (*x*) = cos *x* is defined for every real number.

Let *c *be a real number. Put *x* = *c* + *h*

If *x* ® *c*, then *h* ® 0

*h *(*c*) = cos *c*

Therefore, *h* (*x*) = cos *x* is continuous function.

It can be concluded that,

Therefore, cotangent is continuous except at *x *= *n*p, *n *Î **Z**

Is there an error in this question or solution?

Solution Discuss the Continuity of the Cosine, Cosecant, Secant and Cotangent Functions, Concept: Algebra of Continuous Functions.