# How do you verify tan(x)sin(x) + cos(x)= sec(x) ?

Apr 14, 2016

Recall the following quotient, Pythagorean, and reciprocal identities:

$1. \textcolor{red}{\tan x = \sin \frac{x}{\cos} x}$

$2. \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{{\sin}^{2} x + {\cos}^{2} x = 1}$

$3. \textcolor{b l u e}{\sec x = \frac{1}{\cos} x}$

$1$. To verify the given identity, start by working on the left side. Rewrite $\tan x$ in terms of $\sin x$ and $\cos x$.

Left side:

$\textcolor{red}{\tan} x \sin x + \cos x$

$= \textcolor{red}{\sin \frac{x}{\cos} x} \cdot \sin x + \cos x$

$2$. Simplify.

$= {\sin}^{2} \frac{x}{\cos} x + \cos x$

$= \frac{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{{\sin}^{2} x + {\cos}^{2} x}}{\cos} x$

$3$. Simplify the numerator.

$= \frac{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{1}}{\cos} x$

$= \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \sec x \textcolor{w h i t e}{\frac{a}{a}} |}}}$