Find the Exact Value cos(105 degrees )+cos(15 degrees )

$\cos (105 °) + \cos (15 °)$

Simplify each term.

The exact value of $\cos (105 °)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (75) + \cos (15 °)$

Split $75$ into two angles where the values of the six trigonometric functions are known.

$- \cos (30 + 45) + \cos (15 °)$

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$- (\cos (30) \cos (45) - \sin (30) \sin (45)) + \cos (15 °)$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) + \cos (15 °)$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) + \cos (15 °)$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) + \cos (15 °)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \cos (15 °)$

Simplify $- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

Simplify each term.

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \cos (15 °)$

Combine using the product rule for radicals.

$- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \cos (15 °)$

Multiply $3$ by $2$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \cos (15 °)$

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + \cos (15 °)$

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) + \cos (15 °)$

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) + \cos (15 °)$

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \cos (15 °)$

The exact value of $\cos (15 °)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \cos (45 - 30)$

Separate negation.

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \cos (45 - (30))$

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \cos (45) \cos (30) + \sin (45) \sin (30)$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Combine using the product rule for radicals.

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Multiply $2$ by $3$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6} + \sqrt{2}}{4}$

Combine the numerators over the common denominator.

$\frac{- (\sqrt{6} - \sqrt{2}) + \sqrt{6} + \sqrt{2}}{4}$

Simplify each term.

Apply the distributive property.

$\frac{- \sqrt{6} - - \sqrt{2} + \sqrt{6} + \sqrt{2}}{4}$

Multiply $- - \sqrt{2}$ .

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt{6} + 1 \sqrt{2} + \sqrt{6} + \sqrt{2}}{4}$

Multiply $\sqrt{2}$ by $1$ .

$\frac{- \sqrt{6} + \sqrt{2} + \sqrt{6} + \sqrt{2}}{4}$

Simplify terms.

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$\frac{0 + \sqrt{2} + \sqrt{2}}{4}$

Add $0$ and $\sqrt{2}$ .

$\frac{\sqrt{2} + \sqrt{2}}{4}$

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{2 (\sqrt{2})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \sqrt{2}}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 \sqrt{2}}{2 \cdot 2}$

Rewrite the expression.

$\frac{\sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$

Find the Exact Value cos(105 degrees )+cos(15 degrees )

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