Simplify (tan(a)^2-sin(a)^2)/(tan(a)^2)

$\frac{\tan^{2} (a) - \sin^{2} (a)}{\tan^{2} (a)}$

Step 1

Simplify the numerator.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (a)$ and $b = \sin (a)$ .

$\frac{(\tan (a) + \sin (a)) (\tan (a) - \sin (a))}{\tan^{2} (a)}$

Simplify.

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

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

Factor $\sin (a)$ out of $\frac{\sin (a)}{\cos (a)} + \sin (a)$ .

Factor $\sin (a)$ out of $\frac{\sin (a)}{\cos (a)}$ .

$\frac{(\sin (a) \frac{1}{\cos (a)} + \sin (a)) (\tan (a) - \sin (a))}{\tan^{2} (a)}$

Multiply by $1$ .

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

Factor $\sin (a)$ out of $\sin (a) \frac{1}{\cos (a)} + \sin (a) \cdot 1$ .

$\frac{\sin (a) (\frac{1}{\cos (a)} + 1) (\tan (a) - \sin (a))}{\tan^{2} (a)}$

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

$\frac{\sin (a) (\frac{1}{\cos (a)} + 1) (\frac{\sin (a)}{\cos (a)} - \sin (a))}{\tan^{2} (a)}$

Factor $\sin (a)$ out of $\frac{\sin (a)}{\cos (a)} - \sin (a)$ .

Factor $\sin (a)$ out of $\frac{\sin (a)}{\cos (a)}$ .

$\frac{\sin (a) (\frac{1}{\cos (a)} + 1) (\sin (a) \frac{1}{\cos (a)} - \sin (a))}{\tan^{2} (a)}$

Factor $\sin (a)$ out of $- \sin (a)$ .

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

Factor $\sin (a)$ out of $\sin (a) \frac{1}{\cos (a)} + \sin (a) \cdot - 1$ .

$\frac{\sin (a) (\frac{1}{\cos (a)} + 1) (\sin (a) (\frac{1}{\cos (a)} - 1))}{\tan^{2} (a)}$

$\frac{\sin (a) (\frac{1}{\cos (a)} + 1) \sin (a) (\frac{1}{\cos (a)} - 1)}{\tan^{2} (a)}$

Combine exponents.

Raise $\sin (a)$ to the power of $1$ .

$\frac{\sin^{1} (a) \sin (a) (\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)}{\tan^{2} (a)}$

Raise $\sin (a)$ to the power of $1$ .

$\frac{\sin^{1} (a) \sin^{1} (a) (\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)}{\tan^{2} (a)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sin (a)}^{1 + 1} (\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)}{\tan^{2} (a)}$

Add $1$ and $1$ .

$\frac{\sin^{2} (a) (\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)}{\tan^{2} (a)}$

Step 2

Simplify the denominator.

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

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

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

Factor $\sin^{2} (a)$ out of $\sin^{2} (a) (\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$(\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1) \cos^{2} (a)$

Step 5

Expand $(\frac{1}{\cos (a)} + 1) (\frac{1}{\cos (a)} - 1)$ using the FOIL Method.

Apply the distributive property.

$(\frac{1}{\cos (a)} (\frac{1}{\cos (a)} - 1) + 1 (\frac{1}{\cos (a)} - 1)) \cos^{2} (a)$

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{1}{\cos (a)} \cdot \frac{1}{\cos (a)}$ .

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

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

Raise $\cos (a)$ to the power of $1$ .

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

Raise $\cos (a)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Combine $\frac{1}{\cos (a)}$ and $- 1$ .

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

Move the negative in front of the fraction.

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

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

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

Multiply $- 1$ by $1$ .

$(\frac{1}{\cos^{2} (a)} - \frac{1}{\cos (a)} + \frac{1}{\cos (a)} - 1) \cos^{2} (a)$

Add $- \frac{1}{\cos (a)}$ and $\frac{1}{\cos (a)}$ .

$(\frac{1}{\cos^{2} (a)} + 0 - 1) \cos^{2} (a)$

Add $\frac{1}{\cos^{2} (a)}$ and $0$ .

$(\frac{1}{\cos^{2} (a)} - 1) \cos^{2} (a)$

Step 7

Simplify terms.

Apply the distributive property.

$\frac{1}{\cos^{2} (a)} \cos^{2} (a) - 1 \cos^{2} (a)$

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

Cancel the common factor.

$\frac{1}{\cos^{2} (a)} \cos^{2} (a) - 1 \cos^{2} (a)$

Rewrite the expression.

$1 - 1 \cos^{2} (a)$

Rewrite $- 1 \cos^{2} (a)$ as $- \cos^{2} (a)$ .

$1 - \cos^{2} (a)$

Step 8

Apply pythagorean identity.

$\sin^{2} (a)$

Simplify (tan(a)^2-sin(a)^2)/(tan(a)^2)

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