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# 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

Here, we show you a step-by-step solved example of proving trigonometric identities. This solution was automatically generated by our smart calculator:

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

Starting from the left-hand side (LHS) of the identity

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

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

$L.C.M.=\cos\left(x\right)\left(1+\sin\left(x\right)\right)$
4

Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete

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

Rewrite the sum of fractions as a single fraction with the same denominator

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

When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

$\frac{1+\sin\left(x\right)-\cos\left(x\right)^2}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}$
5

Combine and simplify all terms in the same fraction with common denominator $\cos\left(x\right)\left(1+\sin\left(x\right)\right)$

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

Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2 \frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)} 7 Factor the polynomial \sin\left(x\right)^2+\sin\left(x\right) by it's greatest common factor (GCF): \sin\left(x\right) \frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)} 8 Simplify the fraction \frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)} by \sin\left(x\right)+1 \frac{\sin\left(x\right)}{\cos\left(x\right)} 9 Apply the trigonometric identity: \frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}$$=\tan\left(\theta \right)$

$\tan\left(x\right)$
10

Since we have reached the expression of our goal, we have proven the identity

true

##  Final answer to the problem

true

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