已知tanα=2,求值(2-2sin(α+3π/4)cos(α+π/4))/cos^4α-sin^4α)

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已知tanα=2,求值(2-2sin(α+3π/4)cos(α+π/4))/cos^4α-sin^4α)
已知tanα=2,求值
(2-2sin(α+3π/4)cos(α+π/4))/cos^4α-sin^4α)

已知tanα=2,求值(2-2sin(α+3π/4)cos(α+π/4))/cos^4α-sin^4α)
2-2sin(α+3π/4)cos(α+π/4)
=2-2(sin(2a+π)+sinπ/2)
=2+2sin2a-2
=2sin2a
cos^4α-sin^4α
=(cos^2a+sin^2a)(cos^2a-sin^2a)
=cos^2a-sin^2a
=cos2a
(2-2sin(α+3π/4)cos(α+π/4))/cos^4α-sin^4α)
=2sin2a/cos2a
=2tan2a
=4tana/(1-tan^2a)
=4*2/(1-2^2)
=-8/3

首先sin(α+3π/4)=sin(α+π/4+π/2)=cos(α+π/4),cos^4α-sin^4α=cos^2a-sin^2a
原式={2-2cos^2(a+π/4)}/cos^2a-sin^2a
=(cos^2a+sin^2a+2sinacosa)/(cos^2a-sin^2a)
上下同除cos^2a得(tan^2+1+2tana)/(1-tan^2a)=(4+1+4)/(1-4)=-3