Question

$2{\cos }^2\left(\theta \right)-2{\sin }^2\left(\theta \right)=1,$then $\theta =$

• A. $15°$
• B. $30°$
• C. $45°$
• D. $60°$

Answer 1

The correct option is B

$30°$

Explanation for correct option :

Given, $2{\cos }^2\left(\theta \right)-2{\sin }^2\left(\theta \right)=1,$

$$\begin{gathered} \Rightarrow 2\left(1-{\sin }^2\left(\theta \right)\right)-2{\sin }^2\left(\theta \right)=1\mathbf{[}\mathbf{\because }\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{2}}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}\mathbf{+}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{]} \\ \Rightarrow 2-2{\sin }^2\left(\theta \right)-2{\sin }^2\left(\theta \right)=1 \\ \Rightarrow 2\left(1-2{\sin }^2\left(\theta \right)\right)=1 \\ \Rightarrow \cos \left(2\theta \right)=\frac{1}{2}\left[\mathbf{\because }\mathbf{1}\mathbf{-}\mathbf{2}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{\left(}\mathbf{\theta }\mathbf{\right)}\mathbf{=}\mathbf{cos}\mathbf{\left(}\mathbf{2}\mathbf{\theta }\mathbf{\right)}\right] \\ \Rightarrow \cos \left(2\theta \right)=\cos \left(60°\right) \\ \Rightarrow \theta =30° \end{gathered}$$

Therefore, correct answer is option $\left(B\right)$