# If cos 2x+2 cos x=1 then,

Question:

If $\cos 2 x+2 \cos x=1$ then, $\left(2-\cos ^{2} x\right) \sin ^{2} x$ is equal to

(a) 1

(b) $-1$

(c) $-\sqrt{5}$

(d) $\sqrt{5}$

Solution:

(a) 1

We have,

$\cos 2 x+2 \cos x=1$

$\Rightarrow 2 \cos ^{2} x-1+2 \cos x=1$

$\Rightarrow \cos ^{2} x+\cos x-1=0$

$\Rightarrow \cos x=\frac{-1 \pm \sqrt{1^{2}+4}}{2}$

$\Rightarrow \cos x=\frac{-1 \pm \sqrt{5}}{2}$

$\Rightarrow \cos x=\frac{-1+\sqrt{5}}{2}$

Now,

$\left(2-\cos ^{2} x\right) \sin ^{2} x=\left[2-\left(\frac{-1+\sqrt{5}}{2}\right)^{2}\right]\left(1-\cos ^{2} x\right)$

$=\left[2-\frac{1}{4}(1-2 \sqrt{5}+5)\right]\left(1-\frac{1}{4}(1-2 \sqrt{5}+5)\right)$

$=\frac{1}{4}(1+\sqrt{5})(\sqrt{5}-1)=\frac{4}{4}=1$