$\sin x$, $\cos x$ 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.

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.

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

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

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

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

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

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

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

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

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

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

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

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved