Substitution method

Solve a goniometric equation:


sin4 θ - 1/cos2 θ=cos2 θ - 2

Correct answer:

θ = k*pi

Step-by-step explanation:

sin4 θ − 1/cos2 θ=cos2 θ − 2 (1 − cos2 θ)2 − 1/cos2 θ=cos2 θ − 2 t = cos θ  (1−t2)2 − 1/t2 = t2 −2 t2 ⋅  (1−t2)2 − 1 = t4 −2t2 t6−2t4+t2 − 1  = t4 −2t2 t6−3t4+3t2 − 1  = 0 (t−1)3(t+1)3 = 0  t1​=1 t2​=−1  x1​=arccos(t1​)=arccos1=0 x2​=arccos(t2​)=arccos((−1))≐3.1416 k ∈ Z  θ=x=k⋅ π=k⋅ pi

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