Precalculus Examples

$\tan (x) + \cot (x) = \sec (x) \csc (x)$

Step 1

Subtract $\sec (x) \csc (x)$ from both sides of the equation.

$\tan (x) + \cot (x) - \sec (x) \csc (x) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{\sin (x)}{\cos (x)} + \cot (x) - \sec (x) \csc (x) = 0$

Step 2.1.2

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

$\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)} - \sec (x) \csc (x) = 0$

Step 2.1.3

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

$\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)} - \frac{1}{\cos (x)} \csc (x) = 0$

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

$\sin (x) + \frac{\cos^{1} (x) \cos (x)}{\sin (x)} + \cos (x) (- \frac{1}{\sin (x) \cos (x)}) = \cos (x) \cdot 0$

Step 5.2.3

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

$\sin (x) + \frac{\cos^{1} (x) \cos^{1} (x)}{\sin (x)} + \cos (x) (- \frac{1}{\sin (x) \cos (x)}) = \cos (x) \cdot 0$

Step 5.2.4

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

$\sin (x) + \frac{{\cos (x)}^{1 + 1}}{\sin (x)} + \cos (x) (- \frac{1}{\sin (x) \cos (x)}) = \cos (x) \cdot 0$

Step 5.2.5

Add $1$ and $1$ .

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

Step 5.3

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 9

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

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

Step 10

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 11

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 12

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 13

Apply pythagorean identity.

$\sin (x) + \frac{- \sin^{2} (x)}{\sin (x)} = \cos (x) \cdot 0$

Step 14

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 14.1

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

$\sin (x) + \frac{\sin (x) (- \sin (x))}{\sin (x)} = \cos (x) \cdot 0$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Multiply by $1$ .

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

Step 14.2.2

Cancel the common factor.

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

Step 14.2.3

Rewrite the expression.

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

Step 14.2.4

Divide $- \sin (x)$ by $1$ .

$\sin (x) - \sin (x) = \cos (x) \cdot 0$

Step 15

Subtract $\sin (x)$ from $\sin (x)$ .

$0 = \cos (x) \cdot 0$

Step 16

Multiply $\cos (x)$ by $0$ .

$0 = 0$

Step 17

Since $0 = 0$ , the equation will always be true.

Always true