Question 819982
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
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csc^2(9pi/8)
csc(9&#960;/8)=1/sin(9&#960;/8)
reference angle in quadrant III where csc<0
use sin half-angle formula
sin(9&#960;/8)=sin((9&#960;/4)/2
sin((9&#960;/4)/2=&#8730;[(1-cos(9&#960;/4))/2]=&#8730;[(1-(&#8730;2/2)/2)]=&#8730;[(2-&#8730;2)/4]
sin(9&#960;/8)=&#8730;[(2-&#8730;2)/4]
csc(9&#960;/8)=1/sin(9&#960;/8)=1/&#8730;[(2-&#8730;2)/4]
csc^2(9&#960;/8)=1/(2-&#8730;2)/4=4/(2-&#8730;2)&#8776;6.83
calculator check:
csc^2(9pi/8)=(1/sin(9&#960;/8))^2&#8776;6.83..
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cot^2(-15pi/8)
cot(-15&#960;/8)=1/tan(-15&#960;/8)
reference angle in quadrant II where cot<0
use tan half-angle formula
tan(-15&#960;/8)=tan(-15&#960;/4)/2=[sin(-15&#960;/4)]/[1+cos(-15&#960;/4)]=(&#8730;2/2)/(1+(&#8730;2/2))=&#8730;2/(2+&#8730;2)
cot(-15&#960;/8)=1/tan(-15&#960;/8)=(2+&#8730;2)/&#8730;2
cot^2(-15pi/8)=[(2+&#8730;2)/&#8730;2]^2=5.83
calculator check:
cot^2(-15&#960;/8)=[1/tan(-15&#960;/8)]^2&#8776;5.83..
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cos^2(-4pi/3) + csc^2(9pi/8) + cot^2(-15pi/8)=(1/4)+(4/(2-&#8730;2))+(/(2+&#8730;2)&#8776;.25+6.83+5.83&#8776;12.91