(a) If sin θ+ sin^2 θ=1, then prove that cos^2 θ+ cos^4 θ=1.

\[\begin{aligned} & \sin \theta+\sin ^{2} \theta=1 \\ & \Rightarrow \sin \theta=1-\sin ^{2} \theta=\cos ^{2} \theta \\ & \therefore \cos ^{2} \theta+\cos ^{4} \theta=\cos ^{2} \theta\left(1+\cos ^{2} \theta\right) \\ & =\sin \theta(1+\sin \theta) \\ & =\sin \theta+\sin ^{2} \theta=1 \end{aligned} \]

\[\begin{aligned} & \sec ^{2} \theta=1+\frac{1}{7}=\frac{8}{7} \\ & \cot \theta=\sqrt{7} \Rightarrow \operatorname{cosec}^{2} \theta=1+7=8 \\ & \therefore \text { LHS }=\frac{8-\dfrac{8}{7}}{8+\dfrac{8}{7}}=\frac{48}{64} \\ & \quad=\frac{3}{4}=\text { RHS } \end{aligned} \]