Trigonometry Examples

$\frac{\tan (u - 1)}{\tan (u + 1)} = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 1

Simplify $\frac{\tan (u - 1)}{\tan (u + 1)}$ .

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Step 1.1

Rewrite $\tan (u + 1)$ in terms of sines and cosines.

$\frac{\tan (u - 1)}{\frac{\sin (u + 1)}{\cos (u + 1)}} = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 1.2

Rewrite $\tan (u - 1)$ in terms of sines and cosines.

$\frac{\frac{\sin (u - 1)}{\cos (u - 1)}}{\frac{\sin (u + 1)}{\cos (u + 1)}} = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 1.3

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

$\frac{\sin (u - 1)}{\cos (u - 1)} \cdot \frac{\cos (u + 1)}{\sin (u + 1)} = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{\sin (u - 1)}{\cos (u - 1)}$ to $\tan (u - 1)$ .

$\tan (u - 1) \frac{\cos (u + 1)}{\sin (u + 1)} = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 1.4.2

Convert from $\frac{\cos (u + 1)}{\sin (u + 1)}$ to $\cot (u + 1)$ .

$\tan (u - 1) \cot (u + 1) = \frac{1 - \cot (u)}{1 + \cot (u)}$

Step 2

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

$u \approx 0, - 0.83081186, 0.83081186, - 2.31078078, - 3.14159265, 3.14159265, 3.97240451$

Step 3