What is the value of \(1+\tan^2\alpha+\cot^2\alpha-\sec^2\alpha\,\mathrm{cosec}^2\alpha\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-1\)
The product \(\sec^2\alpha\,\mathrm{cosec}^2\alpha\) is also a sum: \(\sec^2\alpha+\mathrm{cosec}^2\alpha=\frac{\sin^2\alpha+\cos^2\alpha}{\sin^2\alpha\cos^2\alpha}=\sec^2\alpha\,\mathrm{cosec}^2\alpha\). Each term then becomes 1 plus \(\tan^2\) or \(\cot^2\).
\(\sec^2\alpha\,\mathrm{cosec}^2\alpha=\sec^2\alpha+\mathrm{cosec}^2\alpha\)
\(=(1+\tan^2\alpha)+(1+\cot^2\alpha)=2+\tan^2\alpha+\cot^2\alpha\)
Value \(=1+\tan^2\alpha+\cot^2\alpha-2-\tan^2\alpha-\cot^2\alpha=-1\)