Question

f(x)=tanx-cotx g(x)=cot2x+cosec2x find f(pi/8)+g(pi/12)

Answer 1

$\textbf{Given:}$

$f(x)=tanx-cotx$

$g(x)=cot2x+cosec2x$

$\textbf{To find:}$

$f(\frac{\pi}{8})+g(\frac{\pi}{12})$

$\textbf{Solution:}$

$\text{Consider,}$

$f(x)=tanx-cotx$

$f(x)=\dfrac{sinx}{cosx}-\dfrac{cosx}{sinx}$

$f(x)=\dfrac{sin^2x-cos^2x}{cosx\,sinx}$

$f(x)=\dfrac{-2(cos^2x-sin^2x)}{2\,sinx\,cosx}$

$f(x)=\dfrac{-2\,cos2x}{sin2x}$

$f(x)=-2\,cot\,2x$

$g(x)=cot2x+cosec2x$

$\text{Now,}$

$f(\frac{\pi}{8})+g(\frac{\pi}{12})$

$=-2\,cot\,2(\frac{\pi}{8})+cot2(\frac{\pi}{12})+cosec2(\frac{\pi}{12})$

$=-2\,cot\frac{\pi}{4}+cot\frac{\pi}{6}+cosec\frac{\pi}{6}$

$=-2(1)+\sqrt{3}+2$

$=-2+\sqrt{3}+2$

$=\sqrt{3}$

$\therefore\textbf{The value of \bf\,f(\frac{\pi}{8})+g(\frac{\pi}{12}) is \bf\sqrt{3} }$

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