Trigonometry Examples

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

Step 1

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

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Step 1.1

Add parentheses.

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

Step 1.2

Rewrite $- \cot (x) \sin (x)$ in terms of sines and cosines.

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

Step 1.3

Cancel the common factors.

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 3.1.2

Multiply $- \cos (x)$ by $1$ .

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

Step 3.1.3

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

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

Multiply $- \cos (x) \cos (x)$ .

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Step 3.1.5.1

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

$1 - \cos (x) + \cos (x) - (\cos^{1} (x) \cos (x))$

Step 3.1.5.2

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

$1 - \cos (x) + \cos (x) - (\cos^{1} (x) \cos^{1} (x))$

Step 3.1.5.3

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

$1 - \cos (x) + \cos (x) - {\cos (x)}^{1 + 1}$

Step 3.1.5.4

Add $1$ and $1$ .

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

Step 3.2

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

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

Step 3.3

Add $1$ and $0$ .

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

Step 4

Apply pythagorean identity.

$\sin^{2} (x)$