Trigonometry Examples

$\tan (A) = \tan (A) \cdot \csc^{2} (A) + \cot (- A)$

Step 1

Start on the right side.

$\tan (A) \cdot \csc^{2} (A) + \cot (- A)$

Step 2

Since $\cot (- A)$ is an odd function, rewrite $\cot (- A)$ as $- \cot (A)$ .

$\tan (A) \cdot \csc^{2} (A) - \cot (A)$

Step 3

Apply Pythagorean identity in reverse.

$\tan (A) \cdot (1 + \cot^{2} (A)) - \cot (A)$

Step 4

Convert to sines and cosines.

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Step 4.1

Write $\tan (A)$ in sines and cosines using the quotient identity.

$\frac{\sin (A)}{\cos (A)} \cdot (1 + \cot^{2} (A)) - \cot (A)$

Step 4.2

Write $\cot (A)$ in sines and cosines using the quotient identity.

$\frac{\sin (A)}{\cos (A)} \cdot (1 + {(\frac{\cos (A)}{\sin (A)})}^{2}) - \cot (A)$

Step 4.3

Write $\cot (A)$ in sines and cosines using the quotient identity.

$\frac{\sin (A)}{\cos (A)} \cdot (1 + {(\frac{\cos (A)}{\sin (A)})}^{2}) - \frac{\cos (A)}{\sin (A)}$

Step 4.4

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

$\frac{\sin (A)}{\cos (A)} \cdot (1 + \frac{\cos^{2} (A)}{\sin^{2} (A)}) - \frac{\cos (A)}{\sin (A)}$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$\frac{\sin (A)}{\cos (A)} \cdot 1 + \frac{\sin (A)}{\cos (A)} \cdot \frac{{\cos (A)}^{2}}{{\sin (A)}^{2}} - \frac{\cos (A)}{\sin (A)}$

Step 5.1.2

Multiply $\frac{\sin (A)}{\cos (A)}$ by $1$ .

$\frac{\sin (A)}{\cos (A)} + \frac{\sin (A)}{\cos (A)} \cdot \frac{{\cos (A)}^{2}}{{\sin (A)}^{2}} - \frac{\cos (A)}{\sin (A)}$

Step 5.1.3

Combine.

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

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

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

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Step 5.1.4.1.1

Factor $\sin (A)$ out of $\sin (A) {\cos (A)}^{2}$ .

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

Step 5.1.4.1.2

Cancel the common factors.

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Step 5.1.4.1.2.1

Factor $\sin (A)$ out of $\cos (A) {\sin (A)}^{2}$ .

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

Step 5.1.4.1.2.2

Cancel the common factor.

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

Step 5.1.4.1.2.3

Rewrite the expression.

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

Step 5.1.4.2

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

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Step 5.1.4.2.1

Factor $\cos (A)$ out of ${\cos (A)}^{2}$ .

$\frac{\sin (A)}{\cos (A)} + \frac{\cos (A) \cos (A)}{\cos (A) \sin (A)} - \frac{\cos (A)}{\sin (A)}$

Step 5.1.4.2.2

Cancel the common factors.

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Step 5.1.4.2.2.1

Factor $\cos (A)$ out of $\cos (A) \sin (A)$ .

$\frac{\sin (A)}{\cos (A)} + \frac{\cos (A) \cos (A)}{\cos (A) (\sin (A))} - \frac{\cos (A)}{\sin (A)}$

Step 5.1.4.2.2.2

Cancel the common factor.

$\frac{\sin (A)}{\cos (A)} + \frac{\cos (A) \cos (A)}{\cos (A) \sin (A)} - \frac{\cos (A)}{\sin (A)}$

Step 5.1.4.2.2.3

Rewrite the expression.

$\frac{\sin (A)}{\cos (A)} + \frac{\cos (A)}{\sin (A)} - \frac{\cos (A)}{\sin (A)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{\sin (A)}{\cos (A)} + \frac{\cos (A) - \cos (A)}{\sin (A)}$

Step 5.3

Subtract $\cos (A)$ from $\cos (A)$ .

$\frac{\sin (A)}{\cos (A)} + \frac{0}{\sin (A)}$

Step 5.4

Divide $0$ by $\sin (A)$ .

$\frac{\sin (A)}{\cos (A)} + 0$

Step 5.5

Add $\frac{\sin (A)}{\cos (A)}$ and $0$ .

$\frac{\sin (A)}{\cos (A)}$

Step 6

Rewrite $\frac{\sin (A)}{\cos (A)}$ as $\tan (A)$ .

$\tan (A)$

Step 7

Because the two sides have been shown to be equivalent, the equation is an identity.

$\tan (A) = \tan (A) \cdot \csc^{2} (A) + \cot (- A)$ is an identity