Trigonometry Examples

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

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Step 2.1

To write

$\sin (t)$ as a fraction with a common denominator, multiply by

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Multiply

$\sin (t) \sin (t)$ .

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Step 2.3.2.1

Raise

$\sin (t)$ to the power of

$1$ .

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

Step 2.3.2.2

Raise

$\sin (t)$ to the power of

$1$ .

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

Step 2.3.2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.2.4

Add

$1$ and

$1$ .

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

Step 2.3.3

Move

$- 1$ .

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

Step 2.3.4

Reorder

$\sin^{2} (t)$ and

$- 1$ .

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

Step 2.3.5

Rewrite

$- 1$ as

$- 1 (1)$ .

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

Step 2.3.6

Factor

$- 1$ out of

$\sin^{2} (t)$ .

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

Step 2.3.7

Factor

$- 1$ out of

$- 1 (1) - 1 (- \sin^{2} (t))$ .

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

Step 2.3.8

Rewrite

$- 1 (1 - \sin^{2} (t))$ as

$- (1 - \sin^{2} (t))$ .

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

Step 2.3.9

Apply pythagorean identity.

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

Step 2.3.10

Factor

$\cos (t)$ out of

$- \cos^{2} (t) + \sin (t) \cos (t)$ .

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Step 2.3.10.1

Factor

$\cos (t)$ out of

$- \cos^{2} (t)$ .

$\frac{\cos (t) (- \cos (t)) + \sin (t) \cos (t)}{\sin (t) + \cos (t)} + \cos (t)$

Step 2.3.10.2

Factor

$\cos (t)$ out of

$\sin (t) \cos (t)$ .

$\frac{\cos (t) (- \cos (t)) + \cos (t) \sin (t)}{\sin (t) + \cos (t)} + \cos (t)$

Step 2.3.10.3

Factor

$\cos (t)$ out of

$\cos (t) (- \cos (t)) + \cos (t) \sin (t)$ .

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \cos (t)$

Step 2.4

To write

$\cos (t)$ as a fraction with a common denominator, multiply by

$\frac{\sin (t) + \cos (t)}{\sin (t) + \cos (t)}$ .

$\frac{\cos (t) (- \cos (t) + \sin (t))}{\sin (t) + \cos (t)} + \frac{\cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor

$\cos (t)$ out of

$\cos (t) (- \cos (t) + \sin (t)) + \cos (t) (\sin (t) + \cos (t))$ .

$\frac{\cos (t) (- \cos (t) + \sin (t) + \sin (t) + \cos (t))}{\sin (t) + \cos (t)}$

Step 2.6.2

Add

$- \cos (t)$ and

$\cos (t)$ .

$\frac{\cos (t) (0 + \sin (t) + \sin (t))}{\sin (t) + \cos (t)}$

Step 2.6.3

Add

$0$ and

$\sin (t)$ .

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

Step 2.6.4

Add

$\sin (t)$ and

$\sin (t)$ .

$\frac{\cos (t) \cdot 2 \sin (t)}{\sin (t) + \cos (t)}$

Step 2.7

Move

$2$ to the left of

$\cos (t)$ .

$\frac{2 \cos (t) \sin (t)}{\sin (t) + \cos (t)}$

Step 3

Reorder terms.

$\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}$

Step 4

Rewrite

$\frac{2 \cos (t) \sin (t)}{\cos (t) + \sin (t)}$ as

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}$ .

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)}$

Step 5

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

$\frac{2 \sin (t) \cos (t)}{\sin (t) + \cos (t)} = \sin (t) + \cos (t) - \frac{1}{\sin (t) + \cos (t)}$ is an identity