Question

If $\sin A+\cos A=p$ and $\sec A+\cos ecA=q$ , then prove that:
$q(p^2-1)=2p$.

Answer 1

$\sin A+\cos A=p....\left(1\right)$

squaring both sides

${(\sin A+\cos A)}^2=p^2$

${\sin }^{2}A+{\cos }^{2}A+2\sin A\cos A=p^2$ $[\because {\sin }^{2}A+{\cos }^{2}A=1]$

$1+2\sin A\cos A=p^2$

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

$\sec A+cosecA=q$

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

(taking LCM)

$\frac{\sin A+\cos A}{\sin A\cos A}=q$ (from eq. 1 & 2)

$\frac{p}{\frac{p^2-1}{2}}=q$

$2p=q(p^2-1)$

Hence proved