If 2x+2 =1 then, (2- ^2x) ^2x is equal to: - Vidyadip

MCQ

If $\cos2\text{x}+2\cos\text{x}=1$ then, $(2-\cos^2\text{x})\sin^2\text{x}$ is equal to:

✓
$1$
B
$-1$
C
$-\sqrt{5}$
D
$\sqrt{5}$

✓

Answer

Correct option: A.

$1$

We have,
$\Rightarrow2\cos^2\text{x}-1+2\cos\text{x}=1$
$\Rightarrow\cos^2\text{x}+\cos\text{x}-1=0$
$\Rightarrow\cos\text{x}=\frac{-1\pm\sqrt{1^2+4}}{2}$
$\Rightarrow\cos\text{x}=\frac{-1\pm\sqrt{5}}{2}$
$\Rightarrow\cos\text{x}=\frac{-1+\sqrt{5}}{2}$
Now,
$(2-\cos^2\text{x})\sin^2\text{x}=\bigg[2-\Big(\frac{-1+\sqrt{5}}{2}\Big)^2\bigg](1-\cos^2\text{x})$
$=\bigg[2-\frac{1}{4}\Big(1-2\sqrt{5}+5\Big)\bigg]\Big(1-\frac{1}{4}\Big(1-2\sqrt{5}+5\Big)\Big)$
$=\frac{1}{4}\Big(1+\sqrt{5}\Big)\Big(\sqrt{5}-1\Big)$
$=\frac{4}{4}$
$=1$

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