Trigonometry Examples

$2 \tan (x) \csc (x) + 2 \csc (x) + \tan (x) + 1 = 0$

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.

$2 (\tan (x) \csc (x)) + 2 \csc (x) + \tan (x) + 1 = 0$

Step 1.1.2

Rewrite $2 \tan (x) \csc (x)$ in terms of sines and cosines.

$2 (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin (x)}) + 2 \csc (x) + \tan (x) + 1 = 0$

Step 1.1.3

Cancel the common factors.

$2 \frac{1}{\cos (x)} + 2 \csc (x) + \tan (x) + 1 = 0$

Step 1.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$2 \sec (x) + 2 \csc (x) + \tan (x) + 1 = 0$

Step 2

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

$x = \frac{3 π}{4} + π n$ , for any integer $n$

Step 3