Find the Exact Value cos(45 degrees )+cos(135 degrees )-cos(225 degrees )+cos(315 degrees )

$\cos (45 °) + \cos (135 °) - \cos (225 °) + \cos (315 °)$

Step 1

Simplify each term.

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

$\frac{\sqrt{2}}{2} + \cos (135 °) - \cos (225 °) + \cos (315 °)$

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.

$\frac{\sqrt{2}}{2} - \cos (45) - \cos (225 °) + \cos (315 °)$

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

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \cos (225 °) + \cos (315 °)$

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 third quadrant.

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - - \cos (45) + \cos (315 °)$

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

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - - \frac{\sqrt{2}}{2} + \cos (315 °)$

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

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + 1 \frac{\sqrt{2}}{2} + \cos (315 °)$

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

$\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \cos (315 °)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

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

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

Step 2

Simplify terms.

Combine the numerators over the common denominator.

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

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

$\sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$

Find the Exact Value cos(45 degrees )+cos(135 degrees )-cos(225 degrees )+cos(315 degrees )

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