If \(\sin\theta+\mathrm{cosec}\,\theta=4\), what is the value of \(\sin^3\theta+\mathrm{cosec}^3\theta\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 52
\(\sin\theta\) and \(\mathrm{cosec}\,\theta\) are reciprocals, so their product is 1. For two numbers with sum \(s\) and product 1, the identity \(a^3+b^3=(a+b)^3-3ab(a+b)\) gives the sum of cubes at once.
\(\sin\theta\times\mathrm{cosec}\,\theta=1\)
\(\sin^3\theta+\mathrm{cosec}^3\theta=4^3-3\times1\times4\)
\(=64-12=52\)