Question

Prove that

$\frac{\sin A-\cos A}{\sin A+\cos A}=\tan 2A-\sec 2A$

Answer 1

L.H.S

$\frac{\sin A-\cos A}{\sin A+\cos A}$

$\Rightarrow \frac{\sin A-\cos A}{\sin A+\cos A}\times \frac{\sin A-\cos A}{\sin A-\cos A}$

$\Rightarrow \frac{(\sin A-\cos A{)}^2}{{\sin }^{2}A-{\cos }^{2}A}$

$\Rightarrow \frac{{\sin }^{2}A+{\cos }^{2}A-2\sin A\cos A}{-({\cos }^{2}A-{\sin }^{2}A)}$

$\Rightarrow \frac{1-2\sin A\cos A}{-\left(\cos 2A\right)}$

$\Rightarrow \frac{1-\sin 2A}{-\left(\cos 2A\right)}$

$\Rightarrow \frac{\sin 2A-1}{\cos 2A}$

$\Rightarrow \tan 2A-\sec 2A$

R.H.S

Hence, proved.