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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