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)


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