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JEE MainMathematicsModerate

In a Δ\DeltaABC, if cos A cos B + sin A sin B sin C = 1, then
(where a,b,c denotes usual notation)

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1−cosAcosBsinAsinB=sinC≤1⇒cos(A−B)≥1⇒\frac{1 - cosAcosB}{sinAsinB} = sinC \leq 1 \Rightarrow cos(A - B) \geq 1 \Rightarrow
cos(A - B) = 1
∴A=B,So,sinC=1−cos⁡2Asin⁡2A=1⇒c=π2\therefore A = B,So,sinC = \frac{1 - \cos^{2}A}{\sin^{2}A} = 1 \Rightarrow c = \frac{\pi}{2}
∴A=B=π4\therefore A = B = \frac{\pi}{4}
By sine rule we have
a = b and c = 2\sqrt{2} b and c = 2a\sqrt{2}a

Source: Grass-G / M-24 Mathematics Paper 31 Dec.docx

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