Trigonometry Examples

$\cos^{2} (a) \cdot \tan^{2} (a) + \sin^{2} (a) \cdot \cot^{2} (a) = 1$

Step 1

Simplify $\cos^{2} (a) \cdot \tan^{2} (a) + \sin^{2} (a) \cdot \cot^{2} (a)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\cos^{2} (a) \cdot {(\frac{\sin (a)}{\cos (a)})}^{2} + \sin^{2} (a) \cdot \cot^{2} (a) = 1$

Step 1.1.2

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

$\cos^{2} (a) \cdot \frac{\sin^{2} (a)}{\cos^{2} (a)} + \sin^{2} (a) \cdot \cot^{2} (a) = 1$

Step 1.1.3

Cancel the common factor of $\cos^{2} (a)$ .

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Step 1.1.3.1

Cancel the common factor.

$\cos^{2} (a) \cdot \frac{\sin^{2} (a)}{\cos^{2} (a)} + \sin^{2} (a) \cdot \cot^{2} (a) = 1$

Step 1.1.3.2

Rewrite the expression.

$\sin^{2} (a) + \sin^{2} (a) \cdot \cot^{2} (a) = 1$

Step 1.1.4

Rewrite $\cot (a)$ in terms of sines and cosines.

$\sin^{2} (a) + \sin^{2} (a) \cdot {(\frac{\cos (a)}{\sin (a)})}^{2} = 1$

Step 1.1.5

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

$\sin^{2} (a) + \sin^{2} (a) \cdot \frac{\cos^{2} (a)}{\sin^{2} (a)} = 1$

Step 1.1.6

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

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Step 1.1.6.1

Cancel the common factor.

$\sin^{2} (a) + \sin^{2} (a) \cdot \frac{\cos^{2} (a)}{\sin^{2} (a)} = 1$

Step 1.1.6.2

Rewrite the expression.

$\sin^{2} (a) + \cos^{2} (a) = 1$

Step 1.2

Apply pythagorean identity.

$1 = 1$

Step 2

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

$a \approx 0.1, \frac{1}{5}, 0.3, \frac{2}{5}, - \frac{1}{2}, - \frac{3}{5}, - \frac{4}{5}$

Step 3