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Trigonometric Identities Calculator

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1

Solved example of trigonometric identities

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

I. Choose what side of the identity are we going to work on

2

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this particular case, we will choose to work on the left side $\sec\left(x\right)^2+\csc\left(x\right)^2$ to reach the right side $\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$

II. Express in terms of sine and cosine

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

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

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

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

Express both sides of the identity in terms of sine ($\sin(x)$) and cosine ($\cos(x)$)

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

III. Operate, group, simplify

4

Unifying fractions with different denominator

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

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

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

IV. Check if we arrived at the expression we wanted to prove

6

Both expressions are equal

true

Final Answer

true

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