Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Multiply $\frac{\cos (u)}{1 + \sin (u)}$ by $\frac{1 - \sin (u)}{1 - \sin (u)}$ .

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\cos (u)$ by $1$ .

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

Step 4.3

Reorder factors in $\cos (u) + \cos (u) (- \sin (u))$ .

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 + \sin (u)) (1 - \sin (u))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Write $- 1$ as a fraction with denominator $1$ .

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

Step 7

Combine.

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

Step 8

Simplify numerator.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $- (- \cos (u) \sin (u))$ .

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

Step 9

Multiply $1 - {\sin (u)}^{2}$ by $1$ .

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

Step 10

Apply pythagorean identity.

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

Step 11

Simplify.

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Step 11.1

Factor $\cos (u)$ out of $- \cos (u) + \cos (u) \sin (u)$ .

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Step 11.1.1

Factor $\cos (u)$ out of $- \cos (u)$ .

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

Step 11.1.2

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

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

Step 11.1.3

Factor $\cos (u)$ out of $\cos (u) \cdot - 1 + \cos (u) (\sin (u))$ .

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

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

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

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

Step 11.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 11.4

Factor $- 1$ out of $\sin (u)$ .

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

Step 11.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin (u))$ .

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

Step 11.6

Move the negative in front of the fraction.

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

Step 12

Write $- 1$ as a fraction with denominator $1$ .

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

Step 13

Combine.

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

Step 14

Simplify numerator.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Multiply $- 1$ by $1$ .

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

Step 14.3

Multiply $- - \sin (u)$ .

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

Step 15

Multiply $\cos (u)$ by $1$ .

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

Step 16

Simplify.

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Step 16.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 16.2

Factor $- 1$ out of $\sin (u)$ .

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

Step 16.3

Factor $- 1$ out of $- 1 (1) - 1 (- \sin (u))$ .

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

Step 16.4

Move the negative in front of the fraction.

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

Step 17

Now consider the left side of the equation.

$\tan (u) - \sec (u)$

Step 18

Convert to sines and cosines.

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Step 18.1

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

$\frac{\sin (u)}{\cos (u)} - \sec (u)$

Step 18.2

Apply the reciprocal identity to $\sec (u)$ .

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

Step 19

Combine the numerators over the common denominator.

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

Step 20

Simplify.

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Step 20.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 20.2

Factor $- 1$ out of $\sin (u)$ .

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

Step 20.3

Factor $- 1$ out of $- 1 (1) - 1 (- \sin (u))$ .

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

Step 20.4

Move the negative in front of the fraction.

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

Step 21

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

$\tan (u) - \sec (u) = - \frac{\cos (u)}{1 + \sin (u)}$ is an identity