Algebra Examples

tan (θ) + cot (θ) = sec (θ) csc (θ)

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

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

tan (θ)

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

\frac{sin (θ)}{cos (θ)} + cot (θ) = sec (θ) csc (θ)

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

Step 1.1.2

Rewrite

cot (θ)

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = sec (θ) csc (θ)

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = sec (θ) csc (θ)

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = sec (θ) csc (θ)

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

Step 2

Simplify the right side.

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Step 2.1

Simplify

sec (θ) csc (θ)

$\sec (θ) \csc (θ)$ .

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Step 2.1.1

Rewrite

sec (θ)

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = \frac{1}{cos (θ)} csc (θ)

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

Step 2.1.2

Rewrite

csc (θ)

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

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

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

Step 2.1.3

Multiply

\frac{1}{cos (θ)}

$\frac{1}{\cos (θ)}$ by

\frac{1}{sin (θ)}

$\frac{1}{\sin (θ)}$ .

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = \frac{1}{cos (θ) sin (θ)}

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = \frac{1}{cos (θ) sin (θ)}

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

\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)} = \frac{1}{cos (θ) sin (θ)}

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

Step 3

Multiply both sides of the equation by

cos (θ)

$\cos (θ)$ .

cos (θ) (\frac{sin (θ)}{cos (θ)} + \frac{cos (θ)}{sin (θ)}) = cos (θ) \frac{1}{cos (θ) sin (θ)}

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

Step 4

Apply the distributive property.

cos (θ) \frac{sin (θ)}{cos (θ)} + cos (θ) \frac{cos (θ)}{sin (θ)} = cos (θ) \frac{1}{cos (θ) sin (θ)}

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

Step 5

Cancel the common factor of

cos (θ)

$\cos (θ)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Multiply

$\cos (θ) \frac{\cos (θ)}{\sin (θ)}$ .

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Step 6.1

Combine

$\cos (θ)$ and

$\frac{\cos (θ)}{\sin (θ)}$ .

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

Step 6.2

Raise

$\cos (θ)$ to the power of

$1$ .

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

Step 6.3

Raise

$\cos (θ)$ to the power of

$1$ .

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

Step 6.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5

Add

$1$ and

$1$ .

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

Step 7

Cancel the common factor of

$\cos (θ)$ .

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Step 7.1

Factor

$\cos (θ)$ out of

$\cos (θ) \sin (θ)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Subtract

$\frac{1}{\sin (θ)}$ from both sides of the equation.

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

Step 9

Simplify

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

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Step 9.1

Combine the numerators over the common denominator.

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

Step 9.2

Reorder

$\cos^{2} (θ)$ and

$- 1$ .

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

Step 9.3

Rewrite

$- 1$ as

$- 1 (1)$ .

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

Step 9.4

Factor

$- 1$ out of

$\cos^{2} (θ)$ .

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

Step 9.5

Factor

$- 1$ out of

$- 1 (1) - 1 (- \cos^{2} (θ))$ .

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

Step 9.6

Rewrite

$- 1 (1 - \cos^{2} (θ))$ as

$- (1 - \cos^{2} (θ))$ .

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

Step 9.7

Apply pythagorean identity.

$\sin (θ) + \frac{- \sin^{2} (θ)}{\sin (θ)} = 0$

Step 9.8

Cancel the common factor of

$\sin^{2} (θ)$ and

$\sin (θ)$ .

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Step 9.8.1

Factor

$\sin (θ)$ out of

$- \sin^{2} (θ)$ .

$\sin (θ) + \frac{\sin (θ) (- \sin (θ))}{\sin (θ)} = 0$

Step 9.8.2

Cancel the common factors.

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Step 9.8.2.1

Multiply by

$1$ .

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

Step 9.8.2.2

Cancel the common factor.

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

Step 9.8.2.3

Rewrite the expression.

$\sin (θ) + \frac{- \sin (θ)}{1} = 0$

Step 9.8.2.4

Divide

$- \sin (θ)$ by

$1$ .

$\sin (θ) - \sin (θ) = 0$

Step 9.9

Subtract

$\sin (θ)$ from

$\sin (θ)$ .

$0 = 0$

Step 10

Since

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

Always true

Step 11

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$