Question

State whether the following statement is true or false.

$\frac{\sin A}{\sec A+\tan A-1}+\frac{\cos A}{\text{cosec}A+\cot A-1}=1$

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ $L.H.S=\frac{\sin A}{\sec A+\tan A-1}+\frac{cotA}{cosecA+\cot A-1}$

$=\frac{\sin A}{\frac{1}{\cos A}+\frac{\sin A}{\cos A}-1}+\frac{\cos A}{\frac{1}{\sin A}+\frac{\cos A}{\sin A}-1}$

$=\frac{\sin A}{\frac{1+\sin A-\cos A}{\cos A}}+\frac{\cos A}{\frac{1+\cos A-\sin A}{\sin A}}$

$=\frac{\sin A\times \cos A}{1+\sin A-\cos A}+\frac{\sin A\times \cos A}{1+\cos A-\sin A}$

$=\sin A\cos A[\frac{1}{1+\sin A-\cos A}+\frac{1}{1+\cos A-\sin A}]$

$=\sin A\cos A\left[\frac{1+\cos A-\sin A+1+\sin A-\cos A}{(1+\sin A-\cos A)(1+\cos A-\sin A)}\right]$

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

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

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

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

$=\sin A\cos A\times \frac{2}{2\sin A\cos A}$

$=1$

$=R.H.S$