Question

In the photo below please!!!!​

Answer 1

`\cos(\theta) + \sin(\theta)`

Explanation:

Use the sine angle sum identity.

`\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)`

`\sin(45\textdegree + \theta)=\sin(45\textdegree)\cos(\theta)+\cos(45\textdegree)\sin(\theta)`

Now, simplify
`\sin(45\textdegree)` and
`\cos(45\textdegree)` to
`(\sqrt2)/(2)`:

`\sin(45\textdegree + \theta)=(\sqrt2)/(2) \cdot \cos(\theta)+(\sqrt2)/(2) \cdot \sin(\theta)`

Then, factor out
`(\sqrt2)/(2)` from each term.

`\sin(45\textdegree + \theta)=(\sqrt2)/(2) \cdot ( \cos(\theta)+\sin(\theta))`

Finally, identify what is in the parentheses corresponding to the blank space in the question.

`\cos(\theta) + \sin(\theta)`