Question

prove that 2sin² π/6cos²π/3=3/2

prove that..
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Answer 1

Step-by-step explanation:

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Answer 2

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$\huge\bf\red{\underline{\underline{Solution}}}$

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Taking L.H.S

• $\bold{2sin²}$$\dfrac{π}{6}$$\bold{+\:cosec²\:}$$\dfrac{7π}{6}$$\bold{cos²}$$\dfrac{π}{3}$

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putting π = 180°

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= $\bold{2sin²}$$\dfrac{180}{6}$$\bold{+\:cosec²}$$\dfrac{7\:×\:180}{6}$$\bold{cos²}$]$\dfrac{180}{3}$

=$\bold{2sin² \:30° \:+ cosec² \:210°\:cos² \:60}$

=$\bold{2(sin\:30°)²\:+\:(cosec\:210°) \: (cos\:60°)²}$

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Here,

$\bold{sin\:30°\:=}$$\dfrac{1}{2}$& $\bold{sin\:cos\:60°=}$$\dfrac{1}{2}$

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For cosec 210°

⠀⠀⠀⠀⠀⠀let's first calculate sin 210°

⠀⠀⠀⠀⠀⠀sin 210° = $\bold{sin\:(180+30)}$

⠀⠀⠀⠀⠀⠀= $\bold{- \:sin 30°}$

⠀⠀⠀⠀⠀⠀= $\dfrac{-1}{2}$

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So,

cosec 210° = $\dfrac{1}{sin\:210°}$

= $\dfrac{1}{-1}$=$\dfrac{2}{-1}$= - 2

$\text{\large\underline{\red{putting\: values\: in \:equation}}}$

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$\bold{=\:2(sin\:30°)2 \:+ \:(cosec\:210°)? \:(cos\: 60°)²}$

⠀⠀⠀⠀⠀= 2 $\dfrac{1²}{2}$+$\bold{(-2)²}$$\dfrac{1²}{2}$

⠀⠀⠀⠀⠀⠀⠀$\bold{=\:2\:×}$$\dfrac{1}{2}$+ 4 ×$\dfrac{1}{2}$

⠀⠀⠀⠀⠀⠀= $\dfrac{1}{2}$ + 1

⠀⠀⠀⠀⠀⠀= $\dfrac{1}{2}$ + 1

⠀⠀⠀⠀⠀⠀= $\dfrac{3}{2}$

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⠀⠀⠀⠀⠀⠀$\bold{= \:R.H.S}$

⠀⠀⠀⠀⠀⠀$\text{\large\underline{\red{Hence\:proved}}}$

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