Question

Prove that: ${(\sin A+\sec A)}^2+{(\cos A+\text{cosec}A)}^2={(1+\sec A\text{cosec}A)}^2.$

Answer 1

LHS $={(\sin A+\sec A)}^2+{(\cos A+cosecA)}^2$

$={\sin }^{2}A+{\cos }^{2}A+{\sec }^{2}A+cose{c}^{2}A+2\frac{\sin A}{\cos A}+2\frac{\cos A}{\sin A}$

$=1+\left[\frac{1}{{\sin }^{2}A{\cos }^{2}A}\right]+2\left[\frac{1}{\sin A\cos A}\right]$

RHS$={(1+\frac{1}{\sin A\cos A})}^2$

$=1+\frac{1}{{\sin }^{2}A{\cos }^{2}A}+\frac{2}{\sin A\cos A}$

$LHS=RHS$

Hence proved