Let θ , φ ∈ [0, 2 π ] be such that 2cos θ (1 – sin φ ) = sin 2 θ cos φ – 1, tan(2 π – θ ) 0 and …

Question

Let θ , φ ∈ [0, 2 π ] be such that 2cos θ (1 – sin φ ) =  sin 2 θ cos φ – 1, tan(2 π – θ ) > 0 and –1 < sin θ < –  . Then φ cannot satisfy

CGP EDU Academic Team · Accepted Answer

The correct answer is CHECK THE SOLUTION.. (a,c,d)
As tan(2 π – θ ) > 0, – 1 <  sin θ < –, θ ∈ [0, 2 π ]
⇒  < θ < 
Now 2cos θ (1 – sin φ ) = sin 2 θ ( tan θ /2 + cot θ /2)cos φ – 1
⇒ 2cos θ (1 – sin φ ) = 2sin θ cos φ – 1
⇒ 2cos θ + 1 = 2sin( θ + φ )
As θ ∈ ⇒ 2cos θ + 1 ∈ (1, 2) ⇒ 1 < 2sin( θ + φ ) < 2 ⇒ < sin( θ + φ ) < 1
As θ + φ ∈ [0, 4 π ] ⇒ θ + φ ∈  or θ + φ ∈ 
⇒  – θ < φ <  – θ or  – θ < φ <  – θ
⇒ φ ∈ 
∴ correct option is (A, C, D)
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