Find the value of `2sin^(2)pi/6 + cosec^(2) 7pi/6 cos^(2) pi/3?` ``

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dear alex1981, pls. note that cos(pi/3) = 1/2 and not sqrt(3)/2 as you mentioned. 

(sin (pi/3) = sqrt(3)/2).

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Using the identity, cosec x =1/sin x , we get

2 sin^2 (pi/6) + (cos^2 (pi/3)) /(sin^2(7pi/6)) 

using the identity, sin (u+v) = sin u cos v +- cos u sin v

and writing 7pi/6 as (pi + pi/6), we get

2 sin^2(pi/6) + (cos^2(pi/3))/[sin pi cos (pi/6) +- sin (pi/6) cos pi]^2

using the values of sin and cos as sin (pi/6) = 1/2; cos (pi/3) = 1/2; sin pi = 0; cos pi = -1; we get

2 (1/2)^2 + [(1/2)^2]/[-+ 1/2]^2 = 2/4 + (1/4)/(1/4) = 3/2

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