Trigonometry Examples

$(\cos (x) \tan (x) + 1) (\sin (x) - 1)$

Step 1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1

Reorder $\cos (x)$ and $\tan (x)$ .

$(\tan (x) \cos (x) + 1) (\sin (x) - 1)$

Step 1.2

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

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

Step 1.3

Cancel the common factors.

$(\sin (x) + 1) (\sin (x) - 1)$

Step 2

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

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Step 2.1

Apply the distributive property.

$\sin (x) (\sin (x) - 1) + 1 (\sin (x) - 1)$

Step 2.2

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) \cdot - 1 + 1 (\sin (x) - 1)$

Step 2.3

Apply the distributive property.

$\sin (x) \sin (x) + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sin (x) \sin (x)$ .

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Step 3.1.1.1

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

$\sin^{1} (x) \sin (x) + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.1.2

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

$\sin^{1} (x) \sin^{1} (x) + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.1.3

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

${\sin (x)}^{1 + 1} + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.1.4

Add $1$ and $1$ .

$\sin^{2} (x) + \sin (x) \cdot - 1 + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.2

Move $- 1$ to the left of $\sin (x)$ .

$\sin^{2} (x) - 1 \cdot \sin (x) + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.3

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$\sin^{2} (x) - \sin (x) + 1 \sin (x) + 1 \cdot - 1$

Step 3.1.4

Multiply $\sin (x)$ by $1$ .

$\sin^{2} (x) - \sin (x) + \sin (x) + 1 \cdot - 1$

Step 3.1.5

Multiply $- 1$ by $1$ .

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

Step 3.2

Add $- \sin (x)$ and $\sin (x)$ .

$\sin^{2} (x) + 0 - 1$

Step 3.3

Add $\sin^{2} (x)$ and $0$ .

$\sin^{2} (x) - 1$

Step 4

Simplify with factoring out.

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Step 4.1

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 4.2

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

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

Step 4.3

Factor $- 1$ out of $\sin^{2} (x)$ .

$- 1 (1) - 1 (- \sin^{2} (x))$

Step 4.4

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

$- 1 (1 - \sin^{2} (x))$

Step 4.5

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

$- (1 - \sin^{2} (x))$

Step 5

Apply pythagorean identity.

$- \cos^{2} (x)$