Trigonometry Examples

$\cot^{4} (x) + 3 \cot^{2} (x) + 2$

Step 1

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

${(\cot^{2} (x))}^{2} + 3 \cot^{2} (x) + 2$

Step 2

Let $u = \cot^{2} (x)$ . Substitute $u$ for all occurrences of $\cot^{2} (x)$ .

$u^{2} + 3 u + 2$

Step 3

Factor $u^{2} + 3 u + 2$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 3.2

Write the factored form using these integers.

$(u + 1) (u + 2)$

Step 4

Replace all occurrences of $u$ with $\cot^{2} (x)$ .

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

Step 5

Simplify.

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Step 5.1

Rearrange terms.

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

Step 5.2

Apply pythagorean identity.

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