# Proving Trigonometric Identities Calculator

## Get detailed solutions to your math problems with our Proving Trigonometric Identities step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

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### Difficult Problems

1

Solved example of proving trigonometric identities

$\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$

I. Choose what side of the identity to work on

2

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated. In this case, we will choose to work on the left side $\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}$ to reach the right side $\tan\left(x\right)$

II. Express in terms of sine and cosine

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$
3

Express both sides of the identity in terms of sine ($\sin(x)$) and cosine ($\cos(x)$)

$\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$

III. Operate, group, simplify

4

Multiplying the fraction and term

$\frac{1}{\cos\left(x\right)}+\frac{-\cos\left(x\right)}{1+\sin\left(x\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$
5

Unifying fractions with different denominator

$\frac{1+\sin\left(x\right)-\cos\left(x\right)^2}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$
6

Applying the trigonometric identity: $1-\cos\left(\theta\right)^2=\sin\left(\theta\right)^2$

$\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$
7

Factoring by $\sin\left(x\right)$

$\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$
8

Simplify the fraction by $\sin\left(x\right)+1$

$\frac{\sin\left(x\right)}{\cos\left(x\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}$

IV. Check if we arrived at the expression we wanted to prove

9

Both expressions are equal

true