Prove that- sin 2-18-sin 2-9-sin 27 -18-sin 24 -9-2

LHS =sin^2 π/18+sin^2 π/9+sin^2 7 π/18+sin^2 4 π/9 =sin^2 π/18+sin^2 4 π/9+sin^2 π/9+sin^2 7 π/18 =sin^2 ( π/2 4π/9 )+sin^2 4 π/9+sin^2 π/9+sin^2 ( π/2 π/9 ) ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Allied Angles

Prove that: s...

Question

Prove that: $s i n^{2} \frac{π}{18} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18} + s i n^{2} \frac{4 π}{9} = 2$

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Solution

$L H S = s i n^{2} \frac{π}{18} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18} + s i n^{2} \frac{4 π}{9}$

$= s i n^{2} \frac{π}{18} + s i n^{2} \frac{4 π}{9} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18}$

$= s i n^{2} (\frac{π}{2} - \frac{4 π}{9}) + s i n^{2} \frac{4 π}{9} + s i n^{2} \frac{π}{9} + s i n^{2} (\frac{π}{2} - \frac{π}{9}) [∵ \frac{π}{18} = \frac{π}{2} - \frac{4 π}{9} a n d \frac{7 π}{18} = \frac{π}{2} - \frac{π}{9}]$

$= c o s^{2} \frac{4 π}{9} + s i n^{2} \frac{4 π}{9} + s i n^{2} \frac{π}{9} + c o s^{2} \frac{π}{9}$

$(∵ s i n (\frac{π}{2} - θ) = c o s θ) = 1 + 1 [∵ s i n^{2} θ + c o s^{2} θ = 1]$

=2 =RHS Hence proved.

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Similar questions

$s i n^{2} \frac{π}{18} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18} + s i n^{2} 4 \frac{π}{9} =$

The value of $s i n^{2} (\frac{π}{18}) + s i n^{2} (\frac{π}{9}) + s i n^{2} (\frac{7 π}{18}) + s i n^{2} (\frac{4 π}{9})$ is

Q. sin²π/18 + sin²π/9 + sin² 7π/18 + sin² 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0

{sin}^{2} \frac{π}{18} + {sin}^{2} \frac{π}{9} + {sin}^{2} \frac{7 π}{18} + {sin}^{2} \frac{4 π}{9} = k

. Find the value of

k

{sin}^{2} \frac{π}{18} + {sin}^{2} \frac{2 π}{18} + {sin}^{2} \frac{4 π}{18} + {cos}^{2} \frac{π}{18} + {cos}^{2} \frac{2 π}{18} + {cos}^{2} \frac{4 π}{18}

is equal to

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Prove that: sin2π18+sin2π9+sin27π18+sin24π9=2

LHS=sin2π18+sin2π9+sin27π18+sin24π9 =sin2π18+sin24π9+sin2π9+sin27π18 =sin2(π2−4π9)+sin24π9+sin2π9+sin2(π2−π9)[∵π18=π2−4π9 and 7π18=π2−π9] =cos24π9+sin24π9+sin2π9+cos2π9 (∵sin(π2−θ)=cosθ)=1+1[∵sin2θ+cos2θ=1] =2 =RHS Hence proved.

Prove that: $s i n^{2} \frac{π}{18} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18} + s i n^{2} \frac{4 π}{9} = 2$