If tan theta +cot theta = 2 where 0 degree < theta <90 degree, then find value of sin^15theta + cos^45theta?

Question

If #tan theta +cot theta = 2,# where #0^@ < theta <90^@,# then find value of #sin^15theta + cos^45theta?#

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Answer 1 · 2018-03-01T14:11:08

# (sqrt2(2^15+1))/2^23#.

Given that, #tantheta+cottheta=2, (0^@ltthetalt90^@)#.

#:. tantheta+1/tantheta=2#

#:. (tan^2theta+1)/tantheta=2#.

#:. tan^2theta+1=2tantheta, or, #

#:. tan^2theta-2tantheta+1=0#.

#:. (tantheta-1)^2=0#.

#:. tantheta=1, and because, 0^@ltthetalt90^@, #

# theta=45^@#.

#:. sintheta=costheta=1/sqrt2#.

Hence, the reqd. value,

#=sin^15theta+cos^45theta#,

#=(1/sqrt2)^15+(1/sqrt2)^45#,

#=1/sqrt2*1/2^7+1/sqrt2*1/2^22#,

#=1/(sqrt2*2^22)(2^15+1)#.

#:."The Reqd. Value="(sqrt2(2^15+1))/2^23#.