Question

Prove that :
$(\csc A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$

Answer 1

$L.H.S=(cosecA-sinA)(secA-cosA)$

$=(\frac{1}{sinA}-sinA)(\frac{1}{cosA}-cosA)=\frac{(1-si{n}^{2}A)(1-co{s}^{2}A)}{sinAcosA}$

$=\frac{co{s}^{2}A\times si{n}^{2}A}{sinAcosA}=sinAcosA$

R.H.D= $\frac{1}{tanA+cotA}=\frac{1}{\frac{sinA}{cosA}+\frac{cosA}{sinA}}=\frac{sinAcosA}{si{n}^{2}A+co{s}^{2}A}=sinAcosA$

$\Rightarrow L.H.S=R.H.S$