Question

Evaluate: $y={\left(\sin x\right)}^{\tan x}+{\left(\cos x\right)}^{\sec x}$

Answer 1

$y=(\sin x{)}^{\tan x}+(\cos x{)}^{\text{sec}x}$

Let $y_1=(\sin x{)}^{\tan x}⟹\log y_1=\tan x\log \sin x⟹\frac{d{y}_1}{dx}=(\sin x{)}^{\tan x}\left(\log \sin x\right({\text{sec}}^{2}x)+1)$

$y_2=(\cos x{)}^{\text{sec}x}⟹\log y_2=\text{sec}x\log \cos x⟹\frac{d{y}_2}{dx}=(\cos x{)}^{\text{sec}x}(\text{sec}x\tan x\log \cos x-\tan x\text{sec}x)$

$\frac{dy}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}=\left(\sin x{)}^{\tan x}\right(1+{\text{sec}}^{2}x\log \sin x)+(\cos x{)}^{\text{sec}x}\text{sec}x\tan x(\log \cos x-1)$