Trigonometry Examples

$\cot (s) + \tan (s) = \sec (s) \cdot \csc (s)$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{\cos (s)}{\sin (s)} + \tan (s) = \sec (s) \cdot \csc (s)$

Step 1.1.2

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

$\frac{\cos (s)}{\sin (s)} + \frac{\sin (s)}{\cos (s)} = \sec (s) \cdot \csc (s)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\sec (s) \cdot \csc (s)$ .

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Step 2.1.1

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

$\frac{\cos (s)}{\sin (s)} + \frac{\sin (s)}{\cos (s)} = \frac{1}{\cos (s)} \cdot \csc (s)$

Step 2.1.2

Rewrite $\csc (s)$ in terms of sines and cosines.

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

Step 2.1.3

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

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

Step 3

Multiply both sides of the equation by $\sin (s)$ .

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

Factor $\sin (s)$ out of $\cos (s) \sin (s)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Subtract $\frac{1}{\cos (s)}$ from both sides of the equation.

$\cos (s) + \frac{\sin^{2} (s)}{\cos (s)} - \frac{1}{\cos (s)} = 0$

Step 9

Simplify $\cos (s) + \frac{\sin^{2} (s)}{\cos (s)} - \frac{1}{\cos (s)}$ .

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Step 9.1

Combine the numerators over the common denominator.

$\cos (s) + \frac{\sin^{2} (s) - 1}{\cos (s)} = 0$

Step 9.2

Reorder $\sin^{2} (s)$ and $- 1$ .

$\cos (s) + \frac{- 1 + \sin^{2} (s)}{\cos (s)} = 0$

Step 9.3

Rewrite $- 1$ as $- 1 (1)$ .

$\cos (s) + \frac{- 1 (1) + \sin^{2} (s)}{\cos (s)} = 0$

Step 9.4

Factor $- 1$ out of $\sin^{2} (s)$ .

$\cos (s) + \frac{- 1 (1) - 1 (- \sin^{2} (s))}{\cos (s)} = 0$

Step 9.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (s))$ .

$\cos (s) + \frac{- 1 (1 - \sin^{2} (s))}{\cos (s)} = 0$

Step 9.6

Rewrite $- 1 (1 - \sin^{2} (s))$ as $- (1 - \sin^{2} (s))$ .

$\cos (s) + \frac{- (1 - \sin^{2} (s))}{\cos (s)} = 0$

Step 9.7

Apply pythagorean identity.

$\cos (s) + \frac{- \cos^{2} (s)}{\cos (s)} = 0$

Step 9.8

Cancel the common factor of $\cos^{2} (s)$ and $\cos (s)$ .

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Step 9.8.1

Factor $\cos (s)$ out of $- \cos^{2} (s)$ .

$\cos (s) + \frac{\cos (s) (- \cos (s))}{\cos (s)} = 0$

Step 9.8.2

Cancel the common factors.

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Step 9.8.2.1

Multiply by $1$ .

$\cos (s) + \frac{\cos (s) (- \cos (s))}{\cos (s) \cdot 1} = 0$

Step 9.8.2.2

Cancel the common factor.

$\cos (s) + \frac{\cos (s) (- \cos (s))}{\cos (s) \cdot 1} = 0$

Step 9.8.2.3

Rewrite the expression.

$\cos (s) + \frac{- \cos (s)}{1} = 0$

Step 9.8.2.4

Divide $- \cos (s)$ by $1$ .

$\cos (s) - \cos (s) = 0$

Step 9.9

Subtract $\cos (s)$ from $\cos (s)$ .

$0 = 0$

Step 10

Since $0 = 0$ , the equation will always be true for any value of $s$ .

All real numbers

Step 11

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$