Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Rewrite $\frac{\sin^{2} (a)}{\cos^{2} (a)}$ as ${(\frac{\sin (a)}{\cos (a)})}^{2}$ .

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

Step 3

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

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

Step 4

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

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

Step 5

Convert from $\frac{\sin (a)}{\cos (a)}$ to $\tan (a)$ .

$(\tan (a) + \sin (a)) (\tan (a) - \sin (a))$

Step 6

Expand $(\tan (a) + \sin (a)) (\tan (a) - \sin (a))$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\tan (a) (\tan (a) - \sin (a)) + \sin (a) (\tan (a) - \sin (a))$

Step 6.2

Apply the distributive property.

$\tan (a) \tan (a) + \tan (a) (- \sin (a)) + \sin (a) (\tan (a) - \sin (a))$

Step 6.3

Apply the distributive property.

$\tan (a) \tan (a) + \tan (a) (- \sin (a)) + \sin (a) \tan (a) + \sin (a) (- \sin (a))$

Step 7

Simplify terms.

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Step 7.1

Combine the opposite terms in $\tan (a) \tan (a) + \tan (a) (- \sin (a)) + \sin (a) \tan (a) + \sin (a) (- \sin (a))$ .

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Step 7.1.1

Reorder the factors in the terms $\tan (a) (- \sin (a))$ and $\sin (a) \tan (a)$ .

$\tan (a) \tan (a) - \sin (a) \tan (a) + \sin (a) \tan (a) + \sin (a) (- \sin (a))$

Step 7.1.2

Add $- \sin (a) \tan (a)$ and $\sin (a) \tan (a)$ .

$\tan (a) \tan (a) + 0 + \sin (a) (- \sin (a))$

Step 7.1.3

Add $\tan (a) \tan (a)$ and $0$ .

$\tan (a) \tan (a) + \sin (a) (- \sin (a))$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $\tan (a) \tan (a)$ .

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Step 7.2.1.1

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

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

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

Step 7.2.1.4

Add $1$ and $1$ .

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

Step 7.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.2.3

Multiply $- \sin (a) \sin (a)$ .

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

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

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

Step 7.2.3.4

Add $1$ and $1$ .

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