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Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Asked by Sanjay Last Modified
Answer 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, cosecant is continuous except at x = np, n Î Z
Therefore, secant is continuous except at 
Therefore, cotangent is continuous except at x = np, n Î Z
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