# How do you write 2cos^2 5-1 as a single trigonometric function?

Jul 24, 2016

cos10

#### Explanation:

Using the basic $\textcolor{b l u e}{\text{double angle expansion for cosine}}$
We can develop further expansions.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\cos 2 x = {\cos}^{2} x - {\sin}^{2} x} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(A\right)$

along with $\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\sin}^{2} x + {\cos}^{2} x = 1} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(B\right)$

From (B) we can obtain.

${\sin}^{2} x = 1 - {\cos}^{2} x \text{ and } {\cos}^{2} x = 1 - {\sin}^{2} x$

Substitute these in turn into right side of (A)

$\Rightarrow 1 - {\sin}^{2} x - {\sin}^{2} x = 1 - 2 {\sin}^{2} x$

and ${\cos}^{2} x - \left(1 - {\cos}^{2} x\right) = 2 {\cos}^{2} x - 1$

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

Using the identity $\cos 2 x = 2 {\cos}^{2} x - 1$

$\Rightarrow 2 {\cos}^{2} 5 - 1 = \cos \left(2 \times 5\right) = \cos 10$