#cos^6(pi/12)+sin^6(pi/12)=#?

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#cos^6(pi/12)+sin^6(pi/12)=?#

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Answer 1 · 2017-07-17T17:28:01

I used a calculator.
Answer: 0.8125 or #13/16#

Answer 2 · 2017-07-17T17:54:36

# 13/16.#

We have, #cos^2(pi/12)+sin^2(pi/12)=1.#

#:. {cos^2(pi/12)+sin^2(pi/12)}^3=1^3.#

Since, #(a+b)^3=a^3+b^3+3ab(a+b),#

#:.{cos^2(pi/12)}^3+{sin^2(pi/12)}^3#

#+3cos^2(pi/12)*sin^2(pi/12){cos^2(pi/12)+sin^2(pi/12)}=1.#

#:.cos^6(pi/12)+sin^6(pi/12)+{3cos^2(pi/12)*sin^2(pi/12)}(1)=1.#

#:.cos^6(pi/12)+sin^6(pi/12)=1-3cos^2(pi/12)sin^2(pi/12).#

Let us recall that,

#cos^2theta=1/2(1+cos2theta), and, sin^2theta=1/2(1-cos2theta).#

#:.cos^6(pi/12)+sin^6(pi/12)=1-3{1/2(1+cos(2pi/12)}{1/2(1-cos(2pi/12)},#

#=1-3/4(1+cos(pi/6))(1-cos(pi/6)),#

#=1-3/4{1-cos^2(pi/6)},#

#=1-3/4(sin^2(pi/6)),#

#=1-3/4(1/2)^2,#

#=1-3/16,#

#=13/16.#

Enjoy Maths.!