Math Doubts

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.

Latest Math Topics
Latest Math Problems
Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved