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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

Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

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

Combine $\sec\left(x\right)+\frac{-\cos\left(x\right)}{1+\sin\left(x\right)}$ in a single fraction

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

Multiplying polynomials $\sec\left(x\right)$ and $1+\sin\left(x\right)$

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

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

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

Multiplying the fraction by $\sin\left(x\right)$

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

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

$\tan\left(x\right)$
6

Applying the trigonometric identity: $\sin\left(\theta\right)\cdot\sec\left(\theta\right)=\tan\left(\theta\right)$

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

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

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

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

$\frac{-\cos\left(x\right)+\frac{1}{\cos\left(x\right)}+\frac{\sin\left(x\right)}{\cos\left(x\right)}}{1+\sin\left(x\right)}=\tan\left(x\right)$
9

Add fraction's numerators with common denominators: $\frac{1}{\cos\left(x\right)}$ and $\frac{\sin\left(x\right)}{\cos\left(x\right)}$

$\frac{\frac{1+\sin\left(x\right)}{\cos\left(x\right)}-\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$
10

Combine $\frac{1+\sin\left(x\right)}{\cos\left(x\right)}-\cos\left(x\right)$ in a single fraction

$\frac{\frac{1+\sin\left(x\right)-\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)}}{1+\sin\left(x\right)}=\tan\left(x\right)$
11

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

$\frac{\frac{1+\sin\left(x\right)-\cos\left(x\right)^2}{\cos\left(x\right)}}{1+\sin\left(x\right)}=\tan\left(x\right)$
12

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

$\frac{\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)}}{1+\sin\left(x\right)}=\tan\left(x\right)$
13

Divide fractions $\frac{\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)}}{1+\sin\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$
14

Multiplying polynomials $\cos\left(x\right)$ and $1+\sin\left(x\right)$

$\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)+\cos\left(x\right)\sin\left(x\right)}=\tan\left(x\right)$
15

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

$\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)+\cos\left(x\right)\sin\left(x\right)}=\tan\left(x\right)$
16

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

$\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$
17

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

$\frac{\sin\left(x\right)}{\cos\left(x\right)}=\tan\left(x\right)$
18

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

$\tan\left(x\right)=\tan\left(x\right)$
19

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

true

Final Answer

true

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