Question

use the given triangle to evaluate the expression. if necessary, express the value without a square root in the denominator by rationalizing the denominator. cos(π/4) - sin(π/4) cos(π/4) - sin(π/4) = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression.

Explanation:

Step1: Recall trigonometric - ratio values

In a 45 - 45 - 90 triangle, $\cos45^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}$ and $\sin45^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}$. Given a 45 - 45 - 90 triangle with legs of length 1 and hypotenuse of length $\sqrt{2}$, $\cos45^{\circ}=\frac{1}{\sqrt{2}}$ and $\sin45^{\circ}=\frac{1}{\sqrt{2}}$. Also, $\frac{\pi}{4}$ radians is equal to $45^{\circ}$. So, $\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$ and $\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$.

Step2: Substitute values into the expression

Substitute $\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$ and $\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$ into $\cos\frac{\pi}{4}-\sin\frac{\pi}{4}$. We get $\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}$.

Step3: Simplify the expression

$\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} = 0$.

Answer:

$0$