Trigonometry Examples

$\sin^{2} (x) \sec^{2} (x) - \sin^{2} (x)$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $\sin^{2} (x)$ out of $\sin^{2} (x) \sec^{2} (x)$ .

$\sin^{2} (x) (\sec^{2} (x)) - \sin^{2} (x)$

Step 1.2

Factor $\sin^{2} (x)$ out of $- \sin^{2} (x)$ .

$\sin^{2} (x) (\sec^{2} (x)) + \sin^{2} (x) \cdot - 1$

Step 1.3

Factor $\sin^{2} (x)$ out of $\sin^{2} (x) (\sec^{2} (x)) + \sin^{2} (x) \cdot - 1$ .

$\sin^{2} (x) (\sec^{2} (x) - 1)$

Step 2

Apply pythagorean identity.

$\sin^{2} (x) \tan^{2} (x)$

Step 3

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

$\sin^{2} (x) {(\frac{\sin (x)}{\cos (x)})}^{2}$

Step 4

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\sin^{2} (x) \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 5

Multiply $\sin^{2} (x) \frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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Step 5.1

Combine $\sin^{2} (x)$ and $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$\frac{\sin^{2} (x) \sin^{2} (x)}{\cos^{2} (x)}$

Step 5.2

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 5.2.1

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

$\frac{{\sin (x)}^{2 + 2}}{\cos^{2} (x)}$

Step 5.2.2

Add $2$ and $2$ .

$\frac{\sin^{4} (x)}{\cos^{2} (x)}$

Step 6

Factor $\sin^{2} (x)$ out of $\sin^{4} (x)$ .

$\frac{\sin^{2} (x) \sin^{2} (x)}{\cos^{2} (x)}$

Step 7

Multiply by $1$ .

$\frac{\sin^{2} (x) \sin^{2} (x)}{\cos^{2} (x) \cdot 1}$

Step 8

Separate fractions.

$\frac{\sin^{2} (x)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 9

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$\frac{\sin^{2} (x)}{1} \tan^{2} (x)$

Step 10

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) \tan^{2} (x)$