Show that f (x) = tan–1(sin X + cos X) is an increasing function in


Given: f (x) = tan–1(sin X + cos X)


To show: the given function is increasing in .


Explanation: Given f (x) = tan–1(sin X + cos X)


Applying first derivative with respect to x, we get



Applying the differentiation rule for tan-1, we get



Applying the sum rule of differentiation, we get



But the derivative of sin X = cos x and that of cos x = -sin x, so



Expanding (sin x+cos x )2, we get



But sin2x+cos2x = 1 and 2sin Xcos X = sin2x, so the above equation becomes,




Now for f(x) to be decreasing function,


f’(x)≥0



cos x- sin x≥ 0


cos x≥ sin x


But this is possible only if


Hence the given function is increasing function in .


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