Trigonometry Examples

$1 + 2 \cot^{2} (θ) + \cot^{4} (θ) = \csc^{4} (θ)$

Step 1

Start on the left side.

$1 + 2 \cot^{2} (θ) + \cot^{4} (θ)$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$1^{2} + 2 \cot^{2} (θ) + \cot^{4} (θ)$

Step 2.2

Rewrite $\cot^{4} (θ)$ as ${(\cot^{2} (θ))}^{2}$ .

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

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \cot^{2} (θ) = 2 \cdot 1 \cdot \cot^{2} (θ)$

Step 2.4

Rewrite the polynomial.

$1^{2} + 2 \cdot 1 \cdot \cot^{2} (θ) + {(\cot^{2} (θ))}^{2}$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = \cot^{2} (θ)$ .

${(1 + \cot^{2} (θ))}^{2}$

Step 3

Apply pythagorean identity.

${(\csc^{2} (θ))}^{2}$

Step 4

Convert to sines and cosines.

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Step 4.1

Apply the reciprocal identity to $\csc (θ)$ .

${({(\frac{1}{\sin (θ)})}^{2})}^{2}$

Step 4.2

Apply the product rule to $\frac{1}{\sin (θ)}$ .

${(\frac{1^{2}}{\sin^{2} (θ)})}^{2}$

Step 4.3

Apply the product rule to $\frac{1^{2}}{\sin^{2} (θ)}$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply the exponents in ${(1^{2})}^{2}$ .

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Step 5.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1^{2 \cdot 2}}{{({\sin (θ)}^{2})}^{2}}$

Step 5.1.1.2

Multiply $2$ by $2$ .

$\frac{1^{4}}{{({\sin (θ)}^{2})}^{2}}$

Step 5.1.2

One to any power is one.

$\frac{1}{{({\sin (θ)}^{2})}^{2}}$

Step 5.2

Multiply the exponents in ${({\sin (θ)}^{2})}^{2}$ .

$\frac{1}{\sin^{4} (θ)}$

Step 6

Rewrite $\frac{1}{\sin^{4} (θ)}$ as $\csc^{4} (θ)$ .

$\csc^{4} (θ)$

Step 7

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

$1 + 2 \cot^{2} (θ) + \cot^{4} (θ) = \csc^{4} (θ)$ is an identity