SOLUTION: cos^2(-4pi/3) + csc^2(9pi/8) + cot^2(-15pi/8)=?

Question 819982: cos^2(-4pi/3) + csc^2(9pi/8) + cot^2(-15pi/8)=?
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
cos^2(-4pi/3) + csc^2(9pi/8) + cot^2(-15pi/8)=?
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cos^2(-4pi/3)
cos(-4π/3)=-1/2
referenc angle in quadrant II where cos<0
cos^2(-4pi/3)=1/4
..
csc^2(9pi/8)
csc(9π/8)=1/sin(9π/8)
reference angle in quadrant III where csc<0
use sin half-angle formula
sin(9π/8)=sin((9π/4)/2
sin((9π/4)/2=√[(1-cos(9π/4))/2]=√[(1-(√2/2)/2)]=√[(2-√2)/4]
sin(9π/8)=√[(2-√2)/4]
csc(9π/8)=1/sin(9π/8)=1/√[(2-√2)/4]
csc^2(9π/8)=1/(2-√2)/4=4/(2-√2)≈6.83
calculator check:
csc^2(9pi/8)=(1/sin(9π/8))^2≈6.83..
..
cot^2(-15pi/8)
cot(-15π/8)=1/tan(-15π/8)
reference angle in quadrant II where cot<0
use tan half-angle formula
tan(-15π/8)=tan(-15π/4)/2=[sin(-15π/4)]/[1+cos(-15π/4)]=(√2/2)/(1+(√2/2))=√2/(2+√2)
cot(-15π/8)=1/tan(-15π/8)=(2+√2)/√2
cot^2(-15pi/8)=[(2+√2)/√2]^2=5.83
calculator check:
cot^2(-15π/8)=[1/tan(-15π/8)]^2≈5.83..
..
cos^2(-4pi/3) + csc^2(9pi/8) + cot^2(-15pi/8)=(1/4)+(4/(2-√2))+(/(2+√2)≈.25+6.83+5.83≈12.91
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