Question

If \( \alpha = \frac{\pi}{4} \), find \( (\sin \alpha + \sin \beta)^2 + (\cos \alpha + \cos \beta)^2 \).

Hint:

Remember to simplify trigonometric expressions before expanding to make calculations easier.

Answer 1

Step 1: Simplify the expression.
We are given \( \alpha = \frac{\pi}{4} \). Using the identity for trigonometric functions of \( \alpha = \frac{\pi}{4} \), we know that:
\( \sin \left(\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

Step 2: Substitute values of \( \sin \alpha \) and \( \cos \alpha \).
Substituting \( \sin \alpha = \cos \alpha = \frac{1}{\sqrt{2}} \) into the expression, we get:
\[ (\frac{1}{\sqrt{2}} + \sin \beta)^2 + (\frac{1}{\sqrt{2}} + \cos \beta)^2 \]

Step 3: Expand the squares.
\[ \left( \frac{1}{\sqrt{2}} + \sin \beta \right)^2 + \left( \frac{1}{\sqrt{2}} + \cos \beta \right)^2 = \left( \frac{1}{2} + \sin \beta + \frac{1}{\sqrt{2}} \sin \beta \right) + \left( \frac{1}{2} + \cos \beta + \frac{1}{\sqrt{2}} \cos \beta \right) \]