QuestionStep 1: Expand $(\sin 2 a+\cos 2 a)^{2}$

Studdy Solution

STEP 1

1. We are asked to expand the square of the binomial expression $(\sin 2a + \cos 2a)^2$ .
2. The expansion will involve using the distributive property (also known as the FOIL method for binomials).

STEP 2

1. Apply the distributive property to expand $(\sin 2a + \cos 2a)^2$ .
2. Simplify the resulting expression.

STEP 3

Apply the distributive property (FOIL method) to expand the expression:
$(\sin 2a + \cos 2a)^2 = (\sin 2a + \cos 2a)(\sin 2a + \cos 2a)$
This results in:
$= \sin^2 2a + \sin 2a \cos 2a + \cos 2a \sin 2a + \cos^2 2a$
Combine the middle terms:
$= \sin^2 2a + 2 \sin 2a \cos 2a + \cos^2 2a$

STEP 4

Simplify the expression using the Pythagorean identity:
$\sin^2 2a + \cos^2 2a = 1$
Thus, the expression becomes:
$= 1 + 2 \sin 2a \cos 2a$
Recognize the double angle identity for sine:
$2 \sin 2a \cos 2a = \sin 4a$
So, the expanded expression is:
$= 1 + \sin 4a$

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