Question

The expression $\sin A(1+\tan A)+\cos A(1+\cot A)$ is equivalent to

• A. $\sin A+\cos A$
• B. $\sec A+\tan A$
• C. $\text{cosec}A+\cot A$
• D. $\text{cosec}A+\sec A$

Answer 1

The correct option is D $\text{cosec}A+\sec A$

$\sin A(1+\tan A)+\cos A(1+\cot A)$
$=\sin A(1+\frac{\sin A}{\cos A})+\cos A(1+\frac{\cos A}{\sin A})$
$=\frac{\sin A}{\cos A}(\cos A+\sin A)+\frac{\cos A}{\sin A}(\sin A+\cos A)$
$=(\cos A+\sin A)(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A})$
$=(\cos A+\sin A)\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{\sin A\cos A}\right)$
$=\frac{\cos A+\sin A}{\sin A\cos A}$
$=\frac{1}{\sin A}+\frac{1}{\cos A}$
$=\text{cosec}A+\sec A$