Question : Solve for $\theta: \cos ^2 \theta-\sin ^2 \theta=\frac{1}{2}, 0<\theta<90^{\circ}$.
Option 1: 45o
Option 2: 60o
Option 3: 30o
Option 4: 40o
Correct Answer: 30 o
Solution :
Given, cos
2
θ − sin
2
θ = $\frac{1}{2}$
⇒ cos
2
θ = $\frac{1}{2}$ + sin
2
θ
since, cos
2
θ + sin
2
θ = 1
⇒ 1 − sin
2
θ = $\frac{1}{2}$ + sin
2
θ
⇒ sin
2
θ + sin
2
θ = $\frac{1}{2}$
⇒ 2sin
2
θ = $\frac{1}{2}$
⇒ sin
2
θ = $\frac{1}{4}$
⇒ sin θ = ± $\frac{1}{2}$
Since, 0 < θ < 90º, sin θ = $\frac{1}{2}$
⇒ θ = 30º
Hence, the correct answer is 30º.
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