To solve the expression [tex]\( 2 \cos^2\left(45^\circ - \theta\right) \)[/tex], follow these steps:
1. Understand the given expression:
We need to simplify [tex]\( 2 \cos^2\left(45^\circ - \theta\right) \)[/tex].
2. Recall a useful trig identity:
The double-angle identity for cosine is:
[tex]\[
\cos^2 x = \frac{1 + \cos(2x)}{2}
\][/tex]
But here, [tex]\( x = 45^\circ - \theta \)[/tex], so the identity is directly useful. Instead, we use the angle sum and difference formulas:
[tex]\[
\cos(a - b) = \cos a \cos b + \sin a \sin b
\][/tex]
For [tex]\( a = 45^\circ \)[/tex] and [tex]\( b = \theta \)[/tex], we get:
[tex]\[
\cos(45^\circ - \theta) = \cos 45^\circ \cos \theta + \sin 45^\circ \sin \theta
\][/tex]
3. Evaluate trigonometric values at specific angles:
We know:
[tex]\[
\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}
\][/tex]
Hence:
[tex]\[
\cos(45^\circ - \theta) = \frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta
\][/tex]
Factor out [tex]\( \frac{\sqrt{2}}{2} \)[/tex]:
[tex]\[
\cos(45^\circ - \theta) = \frac{\sqrt{2}}{2} (\cos \theta + \sin \theta)
\][/tex]
4. Square the cosine expression:
[tex]\[
\cos^2(45^\circ - \theta) = \left( \frac{\sqrt{2}}{2} (\cos \theta + \sin \theta) \right)^2
\][/tex]
Squaring the term inside the parenthesis:
[tex]\[
\cos^2(45^\circ - \theta) = \left( \frac{\sqrt{2}}{2} \right)^2 (\cos \theta + \sin \theta)^2
\][/tex]
Simplify:
[tex]\[
\frac{2}{4} (\cos \theta + \sin \theta)^2 = \frac{1}{2} (\cos \theta + \sin \theta)^2
\][/tex]
5. Multiply by the outside constant:
Now, we multiply by 2:
[tex]\[
2 \cos^2(45^\circ - \theta) = 2 \cdot \frac{1}{2} (\cos \theta + \sin \theta)^2
\][/tex]
[tex]\[
= (\cos \theta + \sin \theta)^2
\][/tex]
6. Simplify the expression further:
Rewriting [tex]\( \cos \theta + \sin \theta \)[/tex]:
[tex]\[
= (\sin \theta + \cos \theta)^2 = \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta
\][/tex]
Using [tex]\( \sin^2 \theta + \cos^2 \theta = 1 \)[/tex]:
[tex]\[
= 1 + 2 \sin \theta \cos \theta
\][/tex]
Given the structure of trigonometric identities, we recognize in radian measure that this translates:
[tex]\[
2 \sin^2 \left(\theta + \frac{\pi}{4}\right)
\][/tex]
So we have:
[tex]\[ 2 \cos^2\left(45^\circ - \theta\right) = 2 \sin^2 \left(\theta + \frac{\pi}{4}\right) \][/tex]
Thus, the simplified form of [tex]\( 2 \cos^2\left(45^\circ - \theta\right) \)[/tex] is:
[tex]\[ 2 \sin^2\left(\theta + \frac{\pi}{4}\right) \][/tex]
