Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

Add $2$ and $2$ .

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

Step 1.4

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

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

Step 1.5

Multiply by $1$ .

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

Step 1.6

Separate fractions.

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

Step 1.7

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

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

Step 1.8

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

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx 0, - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3$

Decimal Form:

$x \approx 0, - 0.1, 0.1, - 0.2, 0.2, - 0.3, 0.3$

Step 4