Question : If $4 \cos θ + 3 \sin θ = x$ and $4 \sin θ − 3 \cos θ = y$, find the value of $x^2 + y^2$.
Option 1: 16
Option 2: 9
Option 3: 25
Option 4: 1
Correct Answer: 25
Solution :
Given: $(4 \cos θ + 3 \sin θ)^2 = x^2$
$(4 \sin θ - 3 \cos θ)^2 = y^2$
Now, $x^2 + y^2 = (4 \cos θ + 3 \sin θ)^2 + (4 \sin θ - 3 \cos θ)^2$
$= 16 \cos^2 θ + 24 \cos θ \sin θ + 9 \sin^2 θ + 16 \sin^2 θ - 24 \cos θ \sin θ + 9 \cos^2 θ$
$= 16(\cos^2 θ + \sin^2 θ) + 9(\cos^2 θ + \sin^2 θ)$
$ = (16\times1) + (9\times1$) [Since $\cos^2 θ + \sin^2 θ = 1$]
$= 25$
Hence, the correct answer is 25.
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