Trigonometry Examples

$\sin (x) + 3 \tan (x) \cot (x) - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Add parentheses.

$\sin (x) + 3 (\tan (x) \cot (x)) - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1.1.2

Reorder $\tan (x)$ and $\cot (x)$ .

$\sin (x) + 3 (\cot (x) \tan (x)) - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1.1.3

Rewrite $3 \tan (x) \cot (x)$ in terms of sines and cosines.

$\sin (x) + 3 (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}) - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1.1.4

Cancel the common factors.

$\sin (x) + 3 \cdot 1 - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1.2

Multiply $3$ by $1$ .

$\sin (x) + 3 - \frac{1}{\csc (x)} + \tan^{2} (x)$

Step 1.3

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

$\sin (x) + 3 - \frac{1}{\frac{1}{\sin (x)}} + \tan^{2} (x)$

Step 1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\sin (x) + 3 - (1 \sin (x)) + \tan^{2} (x)$

Step 1.5

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

$\sin (x) + 3 - \sin (x) + \tan^{2} (x)$

Step 1.6

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

$\sin (x) + 3 - \sin (x) + {(\frac{\sin (x)}{\cos (x)})}^{2}$

Step 1.7

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

$\sin (x) + 3 - \sin (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 2

Combine the opposite terms in $\sin (x) + 3 - \sin (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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Step 2.1

Subtract $\sin (x)$ from $\sin (x)$ .

$0 + 3 + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 2.2

Add $0$ and $3$ .

$3 + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3

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

$3 + \tan^{2} (x)$