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# Prove the trigonometric identity $\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$

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true

## Step-by-step Solution

Problem to solve:

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

Specify the solving method

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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

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

Learn how to solve trigonometric identities problems step by step online.

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

Learn how to solve trigonometric identities problems step by step online. Prove the trigonometric identity 1/(cos(x)-(cos(x)/(1+sin(x))=tan(x). 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. Obtained the least common multiple, we place the LCM as the denominator of each fraction and in the numerator of each fraction we add the factors that we need to complete. Combine and simplify all terms in the same fraction with common denominator \cos\left(x\right)\left(1+\sin\left(x\right)\right). Apply the trigonometric identity: 1-\cos\left(x\right)^2=\sin\left(x\right)^2.

true

### Explore different ways to solve this problem

Prove from LHS (left-hand side)Prove from RHS (right-hand side)Express everything into Sine and Cosine
$\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$

### Main topic:

Trigonometric Identities

~ 0.08 s