Trigonometry Examples

$(\tan (y) + \cot (y)) \sin (y) \cos (y) = 1$

Step 1

Start on the left side.

$(\tan (y) + \cot (y)) \sin (y) \cos (y)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(\frac{\sin (y)}{\cos (y)} + \cot (y)) \sin (y) \cos (y)$

Step 2.1.2

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

$(\frac{\sin (y)}{\cos (y)} + \frac{\cos (y)}{\sin (y)}) \sin (y) \cos (y)$

Step 2.2

Apply the distributive property.

$(\frac{\sin (y)}{\cos (y)} \sin (y) + \frac{\cos (y)}{\sin (y)} \sin (y)) \cos (y)$

Step 2.3

Multiply $\frac{\sin (y)}{\cos (y)} \sin (y)$ .

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Step 2.3.1

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

$(\frac{\sin (y) \sin (y)}{\cos (y)} + \frac{\cos (y)}{\sin (y)} \sin (y)) \cos (y)$

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.4

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

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

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

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

$\sin^{2} (y) + \cos (y) \cos (y)$

Step 2.7

Multiply $\cos (y) \cos (y)$ .

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

Add $1$ and $1$ .

$\sin^{2} (y) + \cos^{2} (y)$

Step 2.8

Apply pythagorean identity.

$1$

Step 3

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

$(\tan (y) + \cot (y)) \sin (y) \cos (y) = 1$ is an identity