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Prove: $(cosecA-sinA)(secA-cosA)=\frac{1}{tanA+cotA}$.
Given: $(cosecA-sinA)(secA-cosA)=\frac{1}{tanA+cotA}$.
To do: To prove: $(cosecA-sinA)(secA-cosA)=\frac{1}{tanA+cotA}$.
Solution:
$(cosecA-sinA)(secA-cosA)=\frac{1}{tanA+cotA}$
$L.H.S.=(cosecA-sinA)(secA-cosA)$
$=( \frac{1}{sinA}-sinA)-( \frac{1}{cosA}-cosA)$ [$\because cosecA=\frac{1}{sinA}$ and $secA=\frac{1}{cosA}$]
$=( \frac{1-sin^2A}{sinA})( \frac{1-cos^2A}{cosA})$
$=( \frac{cos^2A}{sinA})( \frac{sin^2A}{cosA})$ [$\because 1-sin^2A=cos^2A$ and $1-cos^2A=sin^2A$]
$=sinAcosA$
And $R.H.S.=\frac{1}{tanA+cotA}$
$=\frac{1}{\frac{sinA}{cosA}+\frac{cosA}{sinA}}$ [$\because tanA=\frac{sinA}{cosA}$ and $cotA=\frac{cosA}{sinA}$]
$=\frac{1}{\frac{sin^2A+cos^2}{cosAsinA}}$ [$\because sin^2A+cos^2A=1$]
$=sinAcosA$
Thus, $L.H.S.=R.H.S.$