Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Apply the distributive property.

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

Step 5

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

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

Step 6

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

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Step 6.1

Move the leading negative in $- \frac{\sin^{2} (x)}{\cos (x)}$ into the numerator.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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