Algebra Examples

${(1 + \cot (x))}^{2} = \csc^{2} (x) + 2 \cot (x)$

Step 1

Start on the right side.

$\csc^{2} (x) + 2 \cot (x)$

Step 2

Apply Pythagorean identity in reverse.

$1 + \cot^{2} (x) + 2 \cot (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

Write $\cot (x)$ in sines and cosines using the quotient identity.

$1 + {(\frac{\cos (x)}{\sin (x)})}^{2} + 2 \cot (x)$

Step 3.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3.3

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 4

Simplify.

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Step 4.1

Combine $2$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 4.2

To write $\frac{2 \cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 4.3

Write each expression with a common denominator of ${\sin (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{2 \cos (x)}{\sin (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Add $1$ and $1$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Factor $\cos (x)$ out of $2 \cos (x) \sin (x)$ .

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

Step 4.5.3

Factor $\cos (x)$ out of $\cos (x) \cos (x) + \cos (x) (2 \sin (x))$ .

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

Step 4.6

Write $1$ as a fraction with a common denominator.

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

Step 4.7

Combine the numerators over the common denominator.

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

Step 4.8

Simplify the numerator.

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

Step 5

Rewrite $\frac{{(\sin (x) + \cos (x))}^{2}}{\sin^{2} (x)}$ as ${(1 + \cot (x))}^{2}$ .

${(1 + \cot (x))}^{2}$

Step 6

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

${(1 + \cot (x))}^{2} = \csc^{2} (x) + 2 \cot (x)$ is an identity