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

Question

Philoid · Accepted Answer

We know that if g and h are two continuous functions, then,

(i)

(ii)

(iii)

So, first we have to prove that g(x) = sinx and h(x) = cosx are continuous functions.

Let g(x) = sinx

We know that g(x) = sinx is defined for every real number.

Let h be a real number. Now, put x = k + h

So, if

g(k) = sink

= sinkcos0 + cosksin0

= sink + 0

= sink

Thus,

Therefore, g is a continuous function…………(1)

Let h(x) = cosx

We know that h(x) = cosx is defined for every real number.

Let k be a real number. Now, put x = k + h

So, if

h(k) = sink

= coskcos0 - sinksin0

= cosk - 0

= cosk

Thus,

Therefore, g is a continuous function…………(2)

So, from (1) and (2), we get,

Thus, cosecant is continuous except at x = np, (n ϵ Z)

Thus, secant is continuous except at x = , (n ϵ Z)

Thus, cotangent is continuous except at x = np, (n ϵ Z)