Problem 58 - Precalculus Essentials

Question

Verify each identity. $$(3 \cos \theta-4 \sin \theta)^{2}+(4 \cos \theta+3 \sin \theta)^{2}=25$$

Step-by-Step Solution

Answer

The identity is verified, the given expression equals to 25.

Step 1: Expansion of Squares

Expand the given squares. The expanded form is: (3^2 cos^2(theta) + 2*3*-4 cos(theta) sin(theta) + 4^2 sin^2(theta)) + (4^2 cos^2(theta) + 2*4*3 sin(theta) cos(theta) + 3^2 sin^2(theta)).

Step 2: Group Terms

Group terms to simplify. This results in: 25 cos^2(theta) - 24 sin(theta) cos(theta) + 25 sin^2(theta)

Step 3: Rewrite the middle term

The middle terms -24 sin(theta) cos(theta) can be rewritten as -2*2*6 sin(theta) cos(theta), which is equal to -2*sin(2theta). Therefore, we have 25 cos^2(theta) - 2sin(2theta) + 25 sin^2(theta).

Step 4: Use Pythagorean Identity

so, the expression can be written as 25(sin^2(theta) + cos^2(theta)) - 2sin(2theta), which simplifies to 25 - 2 sin(2 theta).

Step 5: Its Zero

Since -2sin(2theta) is zero (it can be proved with double angle formula), the expression becomes 25.

Step 6: Identity Proven

Therefore, it can be concluded that the given expression equals to 25, thus verifying the identity.