Assertion (A): If 4sin4θ+sin22θ+4cos2π4-θ2=2, then θ lies in 3rd quadrant or 4th quadrant. Reason R: sin2θ=sinθ

(A) Both (A) and (R) are true and (R) is the correct explanation of (A)

(B) Both (A) and (R) are true but (R) is not the correct explanation of (A)

(D) (A) is false but (R) is true

AP EAMCET · Maths · Trigonometric Equations

Assertion $(A) :$ If $\sqrt{4 \sin^{4} θ + \sin^{2} 2 θ} + 4 \cos^{2} (\frac{π}{4} - \frac{θ}{2}) = 2,$ then $θ$ lies in $3^{rd}$ quadrant or $4^{th}$ quadrant.
Reason $(R) : \sqrt{\sin^{2} θ} = \sin θ$

A Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$
B Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A)$
C $(A)$ is true but $(R)$ is false
D $(A)$ is false but $(R)$ is true

Verified Solution

Answer & Solution

Correct Answer

(C) $(A)$ is true but $(R)$ is false

Step-by-step Solution

Detailed explanation

Assertion given:- 4sin4θ+sin22θ+4cos2π2-θ2=2 Simplify the above expression. 4sin4θ+4sin2θcos2θ+21+cos2π2-θ2=2 ⇒4sin2θsin2θ+cos2θ+2{1+sinθ}=2 ⇒4sin2θ+2+2sinθ=2 ⇒|2sinθ|+2sinθ=0…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Assertion (A): If 4sin4θ+sin22θ+4cos2π4-θ2=2, then θ lies in 3rd quadrant or 4th quadrant.Reason R: sin2θ=sinθ

Step-by-step Solution

Assertion $(A) :$ If $\sqrt{4 \sin^{4} θ + \sin^{2} 2 θ} + 4 \cos^{2} (\frac{π}{4} - \frac{θ}{2}) = 2,$ then $θ$ lies in $3^{rd}$ quadrant or $4^{th}$ quadrant.
Reason $(R) : \sqrt{\sin^{2} θ} = \sin θ$