To verify the trigonometric identity:
[tex]\[
\cos \left(45^{\circ}+\theta\right) \cdot \cos \left(45^{\circ}-\theta\right)=\frac{1}{2} \cos 2 \theta
\][/tex]
we can follow these steps:
### Step 1: Express the angles in radians
We know that [tex]\( 45^{\circ} = \frac{\pi}{4} \)[/tex]. So the equation can be rewritten in radians as:
[tex]\[
\cos \left(\frac{\pi}{4}+\theta\right) \cdot \cos \left(\frac{\pi}{4}-\theta\right)=\frac{1}{2} \cos 2 \theta
\][/tex]
### Step 2: Expand the left-hand side using trigonometric identities
We use the product-to-sum identities to expand [tex]\( \cos(A) \cos(B) \)[/tex]:
[tex]\[
\cos(A) \cos(B) = \frac{1}{2} [\cos(A+B) + \cos(A-B)]
\][/tex]
For our equation, let [tex]\( A = \frac{\pi}{4} + \theta \)[/tex] and [tex]\( B = \frac{\pi}{4} - \theta \)[/tex]. Then:
[tex]\[
\cos \left(\frac{\pi}{4}+\theta\right) \cdot \cos \left(\frac{\pi}{4}-\theta\right) = \frac{1}{2} \left[ \cos \left( \left( \frac{\pi}{4} + \theta \right) + \left( \frac{\pi}{4} - \theta \right) \right) + \cos \left( \left( \frac{\pi}{4} + \theta \right) - \left( \frac{\pi}{4} - \theta \right) \right) \right]
\][/tex]
Simplifying the angles inside the cosines:
[tex]\[
\left( \frac{\pi}{4} + \theta \right) + \left( \frac{\pi}{4} - \theta \right) = \frac{\pi}{4} + \theta + \frac{\pi}{4} - \theta = \frac{\pi}{2}
\][/tex]
[tex]\[
\left( \frac{\pi}{4} + \theta \right) - \left( \frac{\pi}{4} - \theta \right) = \frac{\pi}{4} + \theta - \frac{\pi}{4} + \theta = 2\theta
\][/tex]
So, the expression becomes:
[tex]\[
\cos \left(\frac{\pi}{4}+\theta\right) \cdot \cos \left(\frac{\pi}{4}-\theta\right) = \frac{1}{2} \left[ \cos \left( \frac{\pi}{2} \right) + \cos(2\theta) \right]
\][/tex]
We know that [tex]\( \cos \left( \frac{\pi}{2} \right) = 0 \)[/tex], so the equation simplifies to:
[tex]\[
\cos \left(\frac{\pi}{4}+\theta\right) \cdot \cos \left(\frac{\pi}{4}-\theta\right) = \frac{1}{2} \cos(2\theta)
\][/tex]
### Step 3: Compare the left-hand side and the right-hand side
We see that both sides of the equation are equal:
[tex]\[
\cos \left(\frac{\pi}{4}+\theta\right) \cdot \cos \left(\frac{\pi}{4}-\theta\right) = \frac{1}{2} \cos(2\theta)
\][/tex]
Therefore, the trigonometric identity is verified. The original equation holds true.
