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Trigonometric Identities Calculator

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1

Solved example of trigonometric identities

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

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$1+\tan\left(x\right)^2+\csc\left(x\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
3

Apply the trigonometric identity: $\csc\left(x\right)^2$$=1+\cot\left(x\right)^2$

$\tan\left(x\right)^2+2+\cot\left(x\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
4

Recognize that the trinomial $\tan\left(x\right)^2+2+\cot\left(x\right)^2$ is perfect square, so we can rewrite it as $\left(\tan\left(x\right)+\cot\left(x\right)\right)^2$

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

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

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

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

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

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

$\left(\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
8

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

$\left(\frac{1}{\cos\left(x\right)\sin\left(x\right)}\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
9

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{1}{\left(\cos\left(x\right)\sin\left(x\right)\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
10

The power of a product is equal to the product of it's factors raised to the same power

$\frac{1}{\cos\left(x\right)^2\sin\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
11

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

true

Final Answer

true

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