Trigonometry Examples

$\sin^{2} (x) (\csc^{2} (x) + \sec^{2} (x))$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 1.3

One to any power is one.

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

Step 1.4

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

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

Step 1.5

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.6

One to any power is one.

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $\sin^{2} (x)$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Combine $\sin^{2} (x)$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 3

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

$1 + \tan^{2} (x)$

Step 4

Rearrange terms.

$\tan^{2} (x) + 1$

Step 5

Apply pythagorean identity.

$\sec^{2} (x)$