Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

To write $- \frac{\cos (t)}{\sin (t)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cos (t)}{1 - \cos (t)}$ .

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

Step 2.3

To write $\frac{\sin (t)}{1 - \cos (t)}$ as a fraction with a common denominator, multiply by $\frac{\sin (t)}{\sin (t)}$ .

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

Step 2.4

Write each expression with a common denominator of $\sin (t) (1 - \cos (t))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{\cos (t)}{\sin (t)}$ by $\frac{1 - \cos (t)}{1 - \cos (t)}$ .

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

Step 2.4.2

Multiply $\frac{\sin (t)}{1 - \cos (t)}$ by $\frac{\sin (t)}{\sin (t)}$ .

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

Step 2.4.3

Reorder the factors of $(1 - \cos (t)) \sin (t)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Multiply $- 1$ by $1$ .

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

Step 2.6.3

Multiply $- \cos (t) (- \cos (t))$ .

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Step 2.6.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.3.2

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

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

Step 2.6.3.3

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

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

Step 2.6.3.4

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

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

Step 2.6.3.5

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

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

Step 2.6.3.6

Add $1$ and $1$ .

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

Step 2.6.4

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

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.4.4

Add $1$ and $1$ .

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

Step 2.6.5

Rewrite $- \cos (t) + \cos^{2} (t) + \sin^{2} (t)$ in a factored form.

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Step 2.6.5.1

Rearrange terms.

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

Step 2.6.5.2

Apply pythagorean identity.

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

Step 2.7

Cancel the common factor of $- \cos (t) + 1$ and $1 - \cos (t)$ .

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Step 2.7.1

Reorder terms.

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

$\frac{1}{\sin (t)}$

Step 3

Rewrite $\frac{1}{\sin (t)}$ as $\csc (t)$ .

$\csc (t)$

Step 4

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

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