Number of solutions of the equation sin⁡x+cos⁡x−22sin⁡xcos⁡x=0 for x∈[0,π] is

sin⁡x+cos⁡x−22sin⁡xcos⁡x=0⇒ sin⁡x+cos⁡x=22sin⁡xcos⁡x⇒ 1+sin⁡2x=2sin2⁡2x⇒ 2sin2⁡2x−sin⁡2x−1=0⇒ (2sin⁡2x+1)(sin⁡2x−1)=0⇒ sin⁡2x=−1/2 or sin⁡2x=1⇒ 2x=11π/6,π/2⇒ x=11π/12,π/4

Number of solutions of the equation sin⁡x+cos⁡x−22sin⁡xcos⁡x=0 for x∈[0,π] is

sin⁡x+cos⁡x−22sin⁡xcos⁡x=0⇒ sin⁡x+cos⁡x=22sin⁡xcos⁡x⇒ 1+sin⁡2x=2sin2⁡2x⇒ 2sin2⁡2x−sin⁡2x−1=0⇒ (2sin⁡2x+1)(sin⁡2x−1)=0⇒ sin⁡2x=−1/2 or sin⁡2x=1⇒ 2x=11π/6,π/2⇒ x=11π/12,π/4

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Number of solutions of the equation $\sin x + \cos x - 2 \sqrt{2} \sin xcos x = 0 for x \in [0, π]$ is

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Detailed Solution

$\begin{array}{l} \sin x + \cos x - 2 \sqrt{2} \sin xcos x = 0 \\ \Rightarrow \sin x + \cos x = 2 \sqrt{2} \sin xcos x \\ \Rightarrow 1 + \sin 2 x = 2 \sin^{2} 2 x \\ \Rightarrow 2 \sin^{2} 2 x - \sin 2 x - 1 = 0 \\ \Rightarrow (2 \sin 2 x + 1) (\sin 2 x - 1) = 0 \\ \Rightarrow \sin 2 x = - 1 / 2 or \sin 2 x = 1 \\ \Rightarrow 2 x = 11 π / 6, π / 2 \\ \Rightarrow x = 11 π / 12, π / 4 \end{array}$