If \(x=\mathrm{cosec}\,A+\cos A\) and \(y=\mathrm{cosec}\,A-\cos A\), then what is the value of \(\left(\displaystyle\frac{2}{x+y}\right)^2+\left(\displaystyle\frac{x-y}{2}\right)^2-1\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
Adding and subtracting the two given expressions separates \(\mathrm{cosec}\,A\) and \(\cos A\); the expression then becomes \(\sin^2 A+\cos^2 A-1\).
\(x+y=2\,\mathrm{cosec}\,A\), so \(\displaystyle\frac{2}{x+y}=\frac{1}{\mathrm{cosec}\,A}=\sin A\)
\(x-y=2\cos A\), so \(\displaystyle\frac{x-y}{2}=\cos A\)
Value \(=\sin^2 A+\cos^2 A-1=1-1=0\)