Question

(sin⁴θ − cos⁴θ + 1) cosec²θ = ?
(a) 4
(b) 3
(c) 2
(d) 1

Answer 1

(c) 2

$$\begin{gathered} ({\text{sin}}^4\theta -{\text{cos}}^4\theta +1){\text{cosec}}^2\theta \\ \text{=}\{({\text{sin}}^2\theta -{\text{cos}}^2\theta )({\text{sin}}^2\theta +{\text{cos}}^2\theta )+1\}\frac{1}{{\text{sin}}^2\theta } \\ \text{=}[\{({\text{sin}}^2\theta -{\text{cos}}^2\theta )\times 1\}+1]\frac{1}{{\text{sin}}^2\theta } \\ \text{=}\frac{{\text{sin}}^2\theta +(1-{\text{cos}}^2\theta )}{{\text{sin}}^2\theta } \\ \text{=}\frac{{\text{sin}}^2\theta +{\sin }^2\theta }{{\text{sin}}^2\theta }[\because (1-{\text{cos}}^2\theta ={\text{sin}}^2\theta )] \\ \text{=}\frac{2{\text{sin}}^2\theta }{\text{sin}\theta } \\ \text{=}2 \end{gathered}$$