Trigonometry Examples

$\frac{\cos (x) \sin (x)}{\cot (x)} = 1 - \cos^{2} (x)$

Step 1

Start on the left side.

$\frac{\cos (x) \sin (x)}{\cot (x)}$

Step 2

Simplify the expression.

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Step 2.1

Separate fractions.

$\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\cot (x)}$

Step 2.2

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\frac{\cos (x)}{\sin (x)}}$

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.4

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{\sin (x)}{1} (\frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)})$

Step 2.5

Cancel the common factor of $\cos (x)$ .

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Step 2.5.1

Cancel the common factor.

$\frac{\sin (x)}{1} (\frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)})$

Step 2.5.2

Rewrite the expression.

$\frac{\sin (x)}{1} \sin (x)$

Step 2.6

Divide $\sin (x)$ by $1$ .

$\sin (x) \sin (x)$

Step 2.7

Multiply $\sin (x) \sin (x)$ .

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Step 2.7.1

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin (x)$

Step 2.7.2

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin^{1} (x)$

Step 2.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\sin (x)}^{1 + 1}$

Step 2.7.4

Add $1$ and $1$ .

$\sin^{2} (x)$

Step 3

Apply Pythagorean identity in reverse.

$1 - \cos^{2} (x)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\cos (x) \sin (x)}{\cot (x)} = 1 - \cos^{2} (x)$ is an identity