Question

Prove that $\left(\frac{\cos A}{1+\sin A}\right)+\left(\frac{1+\sin A}{\cos A}\right)=2secA$

Answer 1

Consider, $L.H.S=\left(\frac{\cos A}{1+\sin A}\right)+\left(\frac{1+\sin A}{\cos A}\right)$

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

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

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

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

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

$=\frac{2+2\sin A}{\cos A.\left(1+\sin A\right)}$

$=\frac{2(1+\sin A)}{\cos A.\left(1+\sin A\right)}$

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

$=2secA$ [$\frac{1}{\cos A}=secA$]

$=R.H.S.$

Hence it is proved that, $\left(\frac{\cos A}{1+\sin A}\right)+\left(\frac{1+\sin A}{\cos A}\right)=2secA$