Let's solve the problems shown in the image step by step.

Problem (a):

We are given: $\cos^2(33^\circ) - \sin^2(33^\circ) = \cos(A^\circ)$

Solution:

Using the cosine double-angle formula: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

Here, $\theta = 33^\circ$. Thus: $\cos^2(33^\circ) - \sin^2(33^\circ) = \cos(2 \cdot 33^\circ)$

Simplify: $\cos(2 \cdot 33^\circ) = \cos(66^\circ)$

So: $A = 66$

Final Answer for (a):

$A = 66^\circ$

Problem (b):

We are given: $\cos^2(7x) - \sin^2(7x) = \cos(B)$

Solution:

Again using the cosine double-angle formula: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

Here, $\theta = 7x$. Thus: $\cos^2(7x) - \sin^2(7x) = \cos(2 \cdot 7x)$

Simplify: $\cos(2 \cdot 7x) = \cos(14x)$

So: $B = 14x$

Final Answer for (b):

$B = 14x$

Summary:

• (a) $A = 66^\circ$
• (b) $B = 14x$

Would you like more details or examples on similar problems?

Related Questions:

1. What are the other trigonometric double-angle formulas?
2. How do you derive the double-angle formulas from basic trigonometric identities?
3. What happens when using half-angle formulas instead of double-angle formulas?
4. How does the identity $\sin^2(\theta) + \cos^2(\theta) = 1$ connect with this?
5. Can this identity be applied to non-angles (e.g., expressions with $x$)?

Tip:

When solving problems with $\cos^2$ and $\sin^2$, check if double-angle formulas or the Pythagorean identity simplify the expression.