Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

Apply the distributive property.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.4

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

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (t)$ .

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

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

Step 5

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

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