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Proving Trigonometric Identities Calculator

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1

Solved example of proving trigonometric identities

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

Multiplying the fraction by $-1$

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

Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$

$\frac{1+\sin\left(x\right)-\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$
4

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

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

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

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

Factor the polynomial $\sin\left(x\right)^2+\sin\left(x\right)$ by it's GCF: $\sin\left(x\right)$

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

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 $\sin\left(x\right)+1$

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

Apply the trigonometric identity: $\frac{\sin\left(x\right)}{\cos\left(x\right)}$$=\tan\left(x\right)$

$\tan\left(x\right)=\tan\left(x\right)$
9

Since both sides of the equality are equal, we have proven the identity

true

Final Answer

true

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