Questions: Find the exact value of the expression. Do not use a calculator. 1-cos^2 30°-cos^2 45°

Solution Steps

To find the exact value of the expression \(1 - \cos^2 30^\circ - \cos^2 45^\circ\), we can use the Pythagorean identity for cosine, which states that \(\cos^2 \theta = 1 - \sin^2 \theta\). Therefore, we can rewrite the expression using sine values for the given angles. We know that \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\). Substitute these values into the expression and simplify.

We start by using the known values of cosine for the angles given:

• \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 45^\circ = \frac{\sqrt{2}}{2}\)

Substitute these values into the expression \(1 - \cos^2 30^\circ - \cos^2 45^\circ\): \[ 1 - \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{\sqrt{2}}{2}\right)^2 \]

Calculate the squares of the cosine values: \[ \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \] \[ \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} \]

Substitute the squared values back into the expression: \[ 1 - \frac{3}{4} - \frac{1}{2} \]

Convert \(\frac{1}{2}\) to \(\frac{2}{4}\) to have a common denominator: \[ 1 - \frac{3}{4} - \frac{2}{4} = 1 - \frac{5}{4} \]

Simplify the expression: \[ 1 - \frac{5}{4} = \frac{4}{4} - \frac{5}{4} = -\frac{1}{4} \]

Final Answer