Trigonometry Examples

$\frac{\sec (y)}{\tan (y) + \cot (y)} = \sin (y)$

Step 1

Multiply both sides by $\tan (y) + \cot (y)$ .

$\frac{\sec (y)}{\tan (y) + \cot (y)} (\tan (y) + \cot (y)) = \sin (y) (\tan (y) + \cot (y))$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $\tan (y) + \cot (y)$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{\sec (y)}{\tan (y) + \cot (y)} (\tan (y) + \cot (y)) = \sin (y) (\tan (y) + \cot (y))$

Step 2.1.1.2

Rewrite the expression.

$\sec (y) = \sin (y) (\tan (y) + \cot (y))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\sin (y) (\tan (y) + \cot (y))$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

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

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

Step 2.2.1.3.4

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

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

Step 2.2.1.3.5

Add $1$ and $1$ .

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

Step 2.2.1.4

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

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Step 2.2.1.4.1

Cancel the common factor.

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

Step 2.2.1.4.2

Rewrite the expression.

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

Step 2.2.1.5

Simplify each term.

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Step 2.2.1.5.1

Factor $\sin (y)$ out of $\sin^{2} (y)$ .

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

Step 2.2.1.5.2

Separate fractions.

$\sec (y) = \frac{\sin (y)}{1} \cdot \frac{\sin (y)}{\cos (y)} + \cos (y)$

Step 2.2.1.5.3

Convert from $\frac{\sin (y)}{\cos (y)}$ to $\tan (y)$ .

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

Step 2.2.1.5.4

Divide $\sin (y)$ by $1$ .

$\sec (y) = \sin (y) \tan (y) + \cos (y)$

Step 3

Solve for $y$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

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

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

Step 3.2.1.2.3

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

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

Step 3.2.1.2.4

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

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

Step 3.2.1.2.5

Add $1$ and $1$ .

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

Step 3.3

Multiply both sides of the equation by $\cos (y)$ .

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

Step 3.4

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

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

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

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Add $1$ and $1$ .

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

Step 3.8

Apply pythagorean identity.

$1 = 1$

Step 3.9

Since $1 = 1$ , the equation will always be true for any value of $y$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$