Question

已知tan(Θ+π/4）=3,求1+sin4Θ-cos4Θ/1+sin4Θ+cos4Θ的值.

Answer 1

1+sin4Θ-cos4Θ=2sin2θcos2θ+2(sin2θ)^2=2sin2θ(sin2θ+cos2θ)

1+sin4Θ+cos4Θ=2sin2θcos2θ+2(cos2θ)^2=2cos2θ(sin2θ+cos2θ)

∴1+sin4Θ-cos4Θ/1+sin4Θ+cos4Θ

=tan2θ

= - cot2(θ+π/4)

= - (1-tan(θ+π/4)^2)/(2tan(θ+π/4)

=(3^2-1)/(2×3)

=4/3