Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1

Reorder $\sin (x)$ and $\csc (x)$ .

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

Step 1.3.2

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

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

Step 1.3.3

Cancel the common factors.

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

Step 2

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

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

Step 3

Apply pythagorean identity.

$\sec^{2} (x)$