Trigonometry Examples

$\frac{3 \cos^{2} (x) + 4 \cos (x) + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1

Simplify the numerator.

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Step 1.1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$\frac{3 u^{2} + 4 u + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = 4$ .

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Step 1.2.1.1

Factor $4$ out of $4 u$ .

$\frac{3 u^{2} + 4 (u) + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.1.2

Rewrite $4$ as $1$ plus $3$

$\frac{3 u^{2} + (1 + 3) u + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.1.3

Apply the distributive property.

$\frac{3 u^{2} + 1 u + 3 u + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.1.4

Multiply $u$ by $1$ .

$\frac{3 u^{2} + u + 3 u + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{(3 u^{2} + u) + 3 u + 1}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (3 u + 1) + 1 (3 u + 1)}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$\frac{(3 u + 1) (u + 1)}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 1.3

Replace all occurrences of $u$ with $\cos (x)$ .

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{\cos^{2} (x) + 2 \cos (x) + 1}$

Step 2

Simplify the denominator.

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Step 2.1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{u^{2} + 2 u + 1}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{u^{2} + 2 u + 1^{2}}$

Step 2.2.2

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

$2 u = 2 \cdot u \cdot 1$

Step 2.2.3

Rewrite the polynomial.

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{u^{2} + 2 \cdot u \cdot 1 + 1^{2}}$

Step 2.2.4

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

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{{(u + 1)}^{2}}$

Step 2.3

Replace all occurrences of $u$ with $\cos (x)$ .

$\frac{(3 \cos (x) + 1) (\cos (x) + 1)}{{(\cos (x) + 1)}^{2}}$

Step 3

Cancel the common factor of $\cos (x) + 1$ and ${(\cos (x) + 1)}^{2}$ .

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Step 3.1

Factor $\cos (x) + 1$ out of $(3 \cos (x) + 1) (\cos (x) + 1)$ .

$\frac{(\cos (x) + 1) (3 \cos (x) + 1)}{{(\cos (x) + 1)}^{2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\frac{(\cos (x) + 1) (3 \cos (x) + 1)}{(\cos (x) + 1) (\cos (x) + 1)}$

Step 3.2.2

Cancel the common factor.

$\frac{(\cos (x) + 1) (3 \cos (x) + 1)}{(\cos (x) + 1) (\cos (x) + 1)}$

Step 3.2.3

Rewrite the expression.

$\frac{3 \cos (x) + 1}{\cos (x) + 1}$