Question

$\cos \left(15°\right)=$

• A. $\sqrt{\frac{(1+\cos \left(30°\right))}{2}}$
• B. $\sqrt{\frac{(1-\cos \left(30°\right))}{2}}$
• C. $\pm \sqrt{\frac{(1+\cos \left(30°\right))}{2}}$
• D. $\pm \sqrt{\frac{(1-\cos \left(30°\right))}{2}}$

Answer 1

The correct option is A

$\sqrt{\frac{(1+\cos \left(30°\right))}{2}}$

Find the value of $\mathit{c}\mathit{o}\mathit{s}\left(15°\right)$:

As we know,

$\begin{array}{rcl}\cos \left(30°\right)& =& \cos \left(2\times 15°\right)\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\cos \left(30°\right)& =& 2{\cos }^2\left(15°\right)-1\end{array}$ $\left[\because \cos 2\theta =2{\cos }^2\theta -1\right]$

$\Rightarrow$ $\begin{array}{rcl}2{\cos }^2\left(15°\right)& =& 1+\cos \left(30°\right)\end{array}$

$\Rightarrow$ $\begin{array}{rcl}{\cos }^2\left(15°\right)& =& \frac{1+\cos \left(30°\right)}{2}\end{array}$

$\begin{array}{rcl}\mathbf{\therefore }\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\left(15°\right)\mathbf{}& \mathbf{=}& \mathbf{}\sqrt{\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\left(30°\right)}{\mathbf{2}}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\end{array}$

Hence, Option(A) is correct.