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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 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

Final Answer

true

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