Question

prove that 2sin² π/6cos²π/3=3/2

prove that..
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Answer 1

━━━━━━━━━━━━━━━━━━━━━━━━━

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

⠀⠀⠀⠀⠀⠀⠀⠀

Taking L.H.S

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

⠀⠀⠀⠀⠀⠀⠀⠀

putting π = 180°

⠀⠀⠀⠀⠀⠀⠀⠀

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀

Here,

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

⠀⠀⠀⠀

⠀

For cosec 210°

⠀⠀⠀⠀⠀⠀let's first calculate sin 210°

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

⠀⠀⠀⠀⠀⠀= $\bold{- \:sin 30°}$

⠀⠀⠀⠀⠀⠀= $\dfrac{-1}{2}$

⠀⠀⠀⠀⠀

So,

cosec 210° = $\dfrac{1}{sin\:210°}$

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

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

⠀⠀⠀⠀⠀⠀⠀⠀

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

⠀⠀⠀⠀⠀= 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

⠀⠀⠀⠀⠀⠀= $\dfrac{3}{2}$

⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀$\bold{= \:R.H.S}$

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

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Answer 2

Answer:

Taking L.H.S

\bold{2sin²}2sin² \dfrac{π}{6}

6

π

\bold{+\:cosec²\:}+cosec² \dfrac{7π}{6}

6

7π

\bold{cos²}cos² \dfrac{π}{3}

3

π

⠀⠀⠀⠀⠀⠀⠀⠀

putting π = 180°

⠀⠀⠀⠀⠀⠀⠀⠀

= \bold{2sin²}2sin² \dfrac{180}{6}

6

180

\bold{+\:cosec²}+cosec² \dfrac{7\:×\:180}{6}

6

7×180

\bold{cos²}cos² \dfrac{180}{3}

3

180

=\bold{2sin² \:30° \:+ cosec² \:210°\:cos² \:60}2sin²30°+cosec²210°cos²60

=\bold{2(sin\:30°)²\:+\:(cosec\:210°) \: (cos\:60°)²}2(sin30°)²+(cosec210°)(cos60°)²

⠀⠀⠀⠀⠀⠀⠀⠀

Here,

\bold{sin\:30°\:=}sin30°= \dfrac{1}{2}

2

1

& \bold{sin\:cos\:60°=}sincos60°= \dfrac{1}{2}

2

1

⠀⠀⠀⠀

⠀

For cosec 210°

⠀⠀⠀⠀⠀⠀let's first calculate sin 210°

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

⠀⠀⠀⠀⠀⠀= \bold{- \:sin 30°}−sin30°

⠀⠀⠀⠀⠀⠀= \dfrac{-1}{2}

2

−1

⠀⠀⠀⠀⠀

So,

cosec 210° = \dfrac{1}{sin\:210°}

sin210°

1

= \dfrac{1}{-1}

−1

1

=\dfrac{2}{-1}

−1

2

= - 2

\text{\large\underline{\red{putting\: values\: in \:equation}}}2(sin30°)2+(cosec210°)?(cos60°)²

⠀⠀⠀⠀⠀= 2 \dfrac{1²}{2}

2

1²

+\bold{(-2)²}(−2)² \dfrac{1²}{2}

2

1²

⠀⠀⠀⠀⠀⠀⠀\bold{=\:2\:×}=2× \dfrac{1}{2}

2

1

+ 4 ×\dfrac{1}{2}

2

1

⠀⠀⠀⠀⠀⠀= \dfrac{1}{2}

2

1

+ 1

⠀⠀⠀⠀⠀⠀= \dfrac{1}{2}

2

1

+ 1

⠀⠀⠀⠀⠀⠀= \dfrac{3}{2}

2

3

⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀\bold{= \:R.H.S}=R.H.S