# Trigonometric Identities Calculator

## Get detailed solutions to your math problems with our 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

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 identity: $\csc\left(x\right)^2$$=1+\cot\left(x\right)^2 2+\tan\left(x\right)^2+\cot\left(x\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 4 Applying the trigonometric identity: \cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)} 2+\tan\left(x\right)^2+\left(\frac{1}{\tan\left(x\right)}\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 5 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} 2+\tan\left(x\right)^2+\frac{1}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 6 Combine all terms into a single fraction \frac{1+2\tan\left(x\right)^2+\tan\left(x\right)^2\tan\left(x\right)^2}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 7 When multiplying exponents with same base we can add the exponents \frac{1+2\tan\left(x\right)^2+\tan\left(x\right)^{4}}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 8 The trinomial 1+2\tan\left(x\right)^2+\tan\left(x\right)^{4} is a perfect square trinomial, because it's discriminant is equal to zero \Delta=b^2-4ac=2^2-4\left(1\right)\left(1\right) = 0=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 9 Using the perfect square trinomial formula a^2+2ab+b^2=(a+b)^2,\:where\:a=\sqrt{\tan\left(x\right)^{4}}\:and\:b=\sqrt{1}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 10 Factoring the perfect square trinomial \frac{\left(\tan\left(x\right)^{2}+1\right)^{2}}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 11 Applying the trigonometric identity: \tan(x)^2+1=\sec(x)^2 \frac{\sec\left(x\right)^{4}}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 12 Applying the secant identity: \displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)} \frac{\left(\frac{1}{\cos\left(x\right)}\right)^{4}}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 13 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{\frac{1}{\cos\left(x\right)^{4}}}{\tan\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 14 Applying the tangent identity: \displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)} \frac{\frac{1}{\cos\left(x\right)^{4}}}{\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 15 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{\frac{1}{\cos\left(x\right)^{4}}}{\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 16 Simplify the fraction \frac{\cos\left(x\right)^2}{\cos\left(x\right)^{4}\sin\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 17 Applying the cotangent identity: \displaystyle\cot\left(\theta\right)=\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)} \frac{\cot\left(x\right)^2}{\cos\left(x\right)^{4}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 18 Apply the identity: \cot\left(x\right)$$=\frac{\cos\left(x\right)}{\sin\left(x\right)}$

$\frac{\left(\frac{\cos\left(x\right)}{\sin\left(x\right)}\right)^2}{\cos\left(x\right)^{4}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
19

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

$\frac{\frac{\cos\left(x\right)^2}{\sin\left(x\right)^2}}{\cos\left(x\right)^{4}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
20

Since $\cos$ is the reciprocal of $\sec$, $\frac{\frac{\cos\left(x\right)^2}{\sin\left(x\right)^2}}{\cos\left(x\right)^{4}}$ is equivalent to $\frac{\cos\left(x\right)^2}{\sin\left(x\right)^2}\sec\left(x\right)^{4}$

$\frac{\cos\left(x\right)^2}{\sin\left(x\right)^2}\sec\left(x\right)^{4}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
21

Multiplying the fraction by $\sec\left(x\right)^{4}$

$\frac{\cos\left(x\right)^2\sec\left(x\right)^{4}}{\sin\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
22

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

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

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

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

Apply the identity: $\cot\left(x\right)$$=\frac{\cos\left(x\right)}{\sin\left(x\right)} \left(\frac{\cos\left(x\right)}{\sin\left(x\right)}\right)^2\left(\frac{1}{\cos\left(x\right)}\right)^{4} 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{\cos\left(x\right)^2}{\sin\left(x\right)^2}\left(\frac{1}{\cos\left(x\right)}\right)^{4} 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{\cos\left(x\right)^2}{\sin\left(x\right)^2}\frac{1}{\cos\left(x\right)^{4}} Multiplying fractions \frac{\cos\left(x\right)^2}{\sin\left(x\right)^2} \times \frac{1}{\cos\left(x\right)^{4}} \frac{\cos\left(x\right)^2}{\sin\left(x\right)^2\cos\left(x\right)^{4}} Simplify the fraction by \cos\left(x\right) \frac{1}{\sin\left(x\right)^2\cos\left(x\right)^{2}} Split the denominator of \frac{1}{\sin\left(x\right)^2\cos\left(x\right)^{2}} \frac{1}{\sin\left(x\right)^2}\frac{1}{\cos\left(x\right)^{2}} Since \sin is the reciprocal of \csc, \frac{1}{\sin\left(x\right)^2} is equivalent to \csc\left(x\right)^2 \csc\left(x\right)^2\frac{1}{\cos\left(x\right)^{2}} Since \cos is the reciprocal of \sec, \frac{1}{\cos\left(x\right)^{2}} is equivalent to \sec\left(x\right)^{2} \csc\left(x\right)^2\sec\left(x\right)^{2} 23 Rewrite \cot\left(x\right)^2\sec\left(x\right)^{4} in terms of \sec and \csc by applying trigonometric identities \sec\left(x\right)^{2}\csc\left(x\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 24 Applying the cosecant identity: \displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)} \sec\left(x\right)^{2}\left(\frac{1}{\sin\left(x\right)}\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 25 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} \sec\left(x\right)^{2}\frac{1}{\sin\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 26 Multiply the fraction and term \frac{\sec\left(x\right)^{2}}{\sin\left(x\right)^2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 27 Apply the property of the quotient of two powers with the same exponent, inversely: \frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m \left(\frac{\sec\left(x\right)}{\sin\left(x\right)}\right)^{2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2} 28 Apply the identity: \frac{\sec\left(x\right)}{b}$$=\frac{1}{b\cos\left(x\right)}$, where $b=\sin\left(x\right)$

$\left(\frac{1}{\sin\left(x\right)\cos\left(x\right)}\right)^{2}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
29

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

$\frac{1}{\left(\sin\left(x\right)\cos\left(x\right)\right)^{2}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
30

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

$\frac{1}{\sin\left(x\right)^{2}\cos\left(x\right)^{2}}=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$
31

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

true