Trigonometry Examples

$3 \sin^{3} (x) \csc (x) + \cos^{2} (x) + 2 \cos (- x) \cos (x)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$3 \sin^{3} (x) \frac{1}{\sin (x)} + \cos^{2} (x) + 2 \cos (- x) \cos (x)$

Step 1.1.2

Cancel the common factor of $\sin (x)$ .

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Step 1.1.2.1

Factor $\sin (x)$ out of $3 \sin^{3} (x)$ .

$\sin (x) (3 \sin^{2} (x)) \frac{1}{\sin (x)} + \cos^{2} (x) + 2 \cos (- x) \cos (x)$

Step 1.1.2.2

Cancel the common factor.

$\sin (x) (3 \sin^{2} (x)) \frac{1}{\sin (x)} + \cos^{2} (x) + 2 \cos (- x) \cos (x)$

Step 1.1.2.3

Rewrite the expression.

$3 \sin^{2} (x) + \cos^{2} (x) + 2 \cos (- x) \cos (x)$

Step 1.1.3

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$3 \sin^{2} (x) + \cos^{2} (x) + 2 \cos (x) \cos (x)$

Step 1.1.4

Multiply $2 \cos (x) \cos (x)$ .

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Step 1.1.4.1

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

$3 \sin^{2} (x) + \cos^{2} (x) + 2 (\cos^{1} (x) \cos (x))$

Step 1.1.4.2

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

$3 \sin^{2} (x) + \cos^{2} (x) + 2 (\cos^{1} (x) \cos^{1} (x))$

Step 1.1.4.3

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

$3 \sin^{2} (x) + \cos^{2} (x) + 2 {\cos (x)}^{1 + 1}$

Step 1.1.4.4

Add $1$ and $1$ .

$3 \sin^{2} (x) + \cos^{2} (x) + 2 \cos^{2} (x)$

Step 1.2

Simplify terms.

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Step 1.2.1

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

$3 \sin^{2} (x) + 3 \cos^{2} (x)$

Step 1.2.2

Factor $3$ out of $3 \sin^{2} (x)$ .

$3 (\sin^{2} (x)) + 3 \cos^{2} (x)$

Step 1.2.3

Factor $3$ out of $3 \cos^{2} (x)$ .

$3 (\sin^{2} (x)) + 3 (\cos^{2} (x))$

Step 1.2.4

Factor $3$ out of $3 (\sin^{2} (x)) + 3 (\cos^{2} (x))$ .

$3 (\sin^{2} (x) + \cos^{2} (x))$

Step 2

Apply pythagorean identity.

$3 \cdot 1$

Step 3

Multiply $3$ by $1$ .

$3$