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

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

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We, know that if two function g(x) and h(x) are continuous then 
\frac{g(x)}{h(x)} , h(x) \neq0\ is \ continuous\\ \frac{1}{h(x)} , h(x) \neq 0\ is \ continuous\\ \frac{1}{g(x)} , g(x) \neq0\ is \ continuous\\
Lets take g(x) = sin x   and    h(x) = cos x
Let suppose  x = c + h
if  x \rightarrow c , \ then \ h \rightarrow 0
g(c) = \sin c\\ \lim_{x\rightarrow c}g(x) = \lim_{x\rightarrow c}\sin x = \lim_{h\rightarrow 0}\sin (c+h)\\ We \ know \ that\\ \sin(a+b) = \sin a \cos b + \cos a\sin b\\ \lim_{h\rightarrow 0}\sin (c+h) = \lim_{h\rightarrow 0}(\sin c\cos h + \cos c \sin h) = \lim_{h\rightarrow 0}\sin c\cos h + \lim_{h\rightarrow 0}\cos c \sin h
                                                                                                =\sin c\cos 0 + \cos c \sin 0 = \sin c
\lim_{x\rightarrow c}g(x) = g(c)
Hence, function g(x) = \sin x is a continuous function
Now,
h(x) = cos x
Let suppose  x = c + h
if  x \rightarrow c , \ then \ h \rightarrow 0
h(c) = \cos c\\ \lim_{x\rightarrow c}h(x) = \lim_{x\rightarrow c}\cos x = \lim_{h\rightarrow 0}\cos (c+h)\\ We \ know \ that\\ \cos(a+b) = \cos a \cos b + \sin a\sin b\\ \lim_{h\rightarrow 0}\cos (c+h) = \lim_{h\rightarrow 0}(\cos c\cos h + \sin c \sin h) = \lim_{h\rightarrow 0}\cos c\cos h + \lim_{h\rightarrow 0}\sin c \sin h
                                                                                                 =\cos c\cos 0 + \sin c \sin 0 = \cos c
\lim_{x\rightarrow c}h(x) = h(c)
Hence, the function h(x) = \cos x is a continuous function
We proved independently that sin x and cos x is a continous function
So, we can say that
cosec x = \frac{1}{\sin x} = \frac{1}{g(x)}  is also continuous except at x=n\pi
sec x  = \frac{1}{\cos x} = \frac{1}{h(x)}  is also continuous except at x=\frac{(2n+1) \pi}{2}
cot x = \frac{\cos x}{\sin x} = \frac{h(x)}{g(x)}  is also continuous except at x=n\pi

 

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