Question

Prove: $\sec A\left(1-\sin A\right)\left(\sec A+\tan A\right)=1$

Answer 1

Consider the L.H.S.

$=\mathrm{secA}(1-\sin A)(\sec A+\tan A)$

$=(\mathrm{secA}-\mathrm{secA}\sin A)(\sec A+\tan A)$

$=(\mathrm{secA}-\frac{\sin A}{\cos A})(\sec A+\tan A)$

$=(\mathrm{secA}-\tan A)(\sec A+\tan A)$

$={\sec }^{2}A-{\tan }^{2}A$

We know that

${\sec }^2-{\tan }^{2}A=1$

Therefore,

$\Rightarrow {\sec }^{2}A-{\tan }^{2}A=1$

Hence proved.