Answer to Question #262425 in Trigonometry for Shah

Question

Prove that

(a) (sin x + cos x) (tan x + cot x) = sec x + cosec x.

(b) sec x sin x / tan x + cot x = sin² x .

Answer 1

Explanations & Calculations

$\qquad\qquad \begin{aligned} \small (\sin x + \cos x)(\tan x+\cot x)&=\small (\sin x+\cos x)\Big(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\Big)\\ &=\small (\sin x+\cos x)\Big(\frac{\sin^2 x+\cos^2 x}{\sin x \cos x}\Big)\\ &=\small \frac{1}{\cos x}+\frac{1}{\sin x}\\ &=\small \sec x+ \cosec x\\ \\ \\ \small \frac{\sec x \sin x}{\tan x+ \cot x}&=\small \frac{\tan x}{\tan x+\large\frac{1}{\tan x}}\\ &=\small \frac{\tan^2 x}{\tan^2 x+1}\\ &=\small \frac{\tan^2 x}{\sec^2 x}\\ &=\small \frac{\sin^2x}{\cos^2x\times\sec^2x}\\ &=\small \sin^2x \end{aligned}$