Question

Prove ${(\sin x+\csc x)}^2+{(\cos x+\sec x)}^2={\tan }^{2}x+{\cot }^{2}x+7$

Answer 1

${\left(\sin x+\cos ecx\right)}^2+{\left(\cos x+\sec x\right)}^2$

$={\sin }^{2}x+\cos e{c}^{2}x+2\sin x\times \cos ecx+{\cos }^{2}x+{\sec }^{2}x+2\cos x\sec x$

$={\sin }^{2}x+{\cos }^{2}x+\cos e{c}^{2}x+{\sec }^{2}x+2\sin x\times \frac{1}{\sin x}+2\cos x\times \frac{1}{\cos x}$

$=1+1+{\cot }^{2}x+1+{\tan }^{2}x+4$

$=7+{\tan }^{2}x+{\cot }^{2}x$

LHS = RHS