If \(\sin\theta+\text{cosec}\,\theta=2\), then what is the value of \(\sin^2\theta+\text{cosec}^2\theta\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
\(\sin\theta\) and \(\text{cosec}\,\theta\) are reciprocals, so their product is 1. Squaring the given sum brings in the two squares asked for, and the middle term is twice that product.
Squaring the given sum: \((\sin\theta+\text{cosec}\,\theta)^2=2^2=4\)
The square written out: \(\sin^2\theta+\text{cosec}^2\theta+2\sin\theta\,\text{cosec}\,\theta=4\)
The product \(\sin\theta\,\text{cosec}\,\theta\) is 1, so \(\sin^2\theta+\text{cosec}^2\theta=4-2=2\)