Trigonometry Examples

$(\sin (x)) (\tan (x) \cos (x) - \cot (x) \cos (x)) = 1 - 2 \cos^{2} (x)$

Step 1

Start on the left side.

$(\sin (x)) (\tan (x) \cos (x) - \cot (x) \cos (x))$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

$\sin (x) (\frac{\sin (x)}{\cos (x)} \cos (x) - \cot (x) \cos (x))$

Step 2.1.1.2

Cancel the common factors.

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $- \frac{\cos (x)}{\sin (x)} \cos (x)$ .

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Step 2.1.3.1

Combine $\cos (x)$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

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

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Step 2.5.1

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Step 2.5.2

Cancel the common factor.

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

Step 2.5.3

Rewrite the expression.

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Subtract ${\cos (x)}^{2}$ from $- {\cos (x)}^{2}$ .

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

Step 5

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

$(\sin (x)) (\tan (x) \cos (x) - \cot (x) \cos (x)) = 1 - 2 \cos^{2} (x)$ is an identity