Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

We have, $\sin \theta (1+{\sin }^2\theta )=1-{\sin }^2\theta$

$$\Rightarrow \sin \theta (2-{\cos }^2\theta )={\cos }^2\theta$$

Squaring both sides, we get:

$${\sin }^2\theta (2-{\cos }^2\theta {)}^2={\cos }^4\theta$$

$$\begin{array}{rl}& \Rightarrow (1-{\cos }^2\theta )(4-4{\cos }^2\theta +{\cos }^4\theta )={\cos }^4\theta \\ & \Rightarrow -{\cos }^6\theta +5{\cos }^4\theta -8{\cos }^2\theta +4={\cos }^4\theta \\ & \therefore {\cos }^6\theta -4{\cos }^4\theta +8{\cos }^2\theta -4=0\end{array}$$