Question #69254

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Answer 1 · 2018-02-19T15:56:18

#(2+sqrt(2))/4#

We have:

#sin^2(67.5^@)#

Using the identity, #sin(x)=cos(90^@-x)#

#:.sin(67.5^@)=cos(90^@-67.5^@)#

#=cos(22.5^@)#

We got: #cos(22.5^@)=cos(45^@/2)#

Using the half-angle identity, #cos(x/2)=sqrt((cos(x)+1)/2)#

#:.cos(45^@/2)=sqrt((cos(45^@)+1)/2)#

We know that #cos(45^@)=sqrt(2)/2#

#<=>cos(45^@/2)=sqrt((sqrt(2)/2+1)/2)#

#=sqrt((2+sqrt(2))/4)#

#=sqrt(2+sqrt(2))/sqrt(4)#

#=sqrt(2+sqrt(2))/2#

#:.cos(22.5^@)=sqrt(2+sqrt(2))/2#

#<=>sin(67.5^@)=sqrt(2+sqrt(2))/2#

#:.sin^2(67.5^@)=(sqrt(2+sqrt(2))/2)^2#

#=(2+sqrt(2))/4#