Simplify the trigonometric expression $\sin x \tan x + \cos x$

$\sin x$, $\cos x$ and $\tan x$ are trigonometric functions if $x$ is the angle of a right angled triangle. The three trigonometric functions involve in a mathematical relation. The mathematical trigonometric relation is, the summation of the $\cos x$ and product of the $\sin x$ and $\tan x$.

$\sin x \tan x + \cos x$

The trigonometric problem can be solved by simplifying the expression by applying some trigonometric identities.

Step: 1

The expression contains $\sin x$ as a factor in first term and $\cos x$ as the second term. Convert the $\tan x$ term in terms of $\sin x$ and $\cos x$ functions and it is possible by the quotient trigonometry rule of sine and cosine.

$= \sin x \Bigg(\dfrac{\sin x}{\cos x}\Bigg) + \cos x$

Now, simplify the trigonometric expression.

$= \dfrac{\sin x}{1} \times \dfrac{\sin x}{\cos x} + \cos x$

$= \dfrac{\sin x \times \sin x}{1 \times \cos x} + \cos x$

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

$= \dfrac{\sin^2 x}{\cos x} + \dfrac{\cos x}{1}$

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

Step: 2

Apply the Pythagorean trigonometric identity of sine and cosine to express sum of squares of sine and cosine functions. Hence, $\sin^2 x + \cos^2 x = 1$.

$\implies \sin x \tan x + \cos x = \dfrac{1}{\cos x}$

Step: 3

Use the reciprocal property between cosine and secant function to express reciprocal of cosine as secant function in trigonometry.

$\therefore \,\,\,\,\,\, \sin x \tan x + \cos x = \sec x$

It is proved that the sum of $\cos x$ and product of $\sin x$ and $\tan x$ is equal to $\sec x$ and it is the required solution for this trigonometric problem.