Substitution method

Solve goniometric equation:


sin4 θ - 1/cos2 θ=cos2 θ - 2

Result

θ = (Correct answer is: k*pi) Wrong answer

Solution:

sin4θ1/cos2θ=cos2θ2 (1cos2θ)21/cos2θ=cos2θ2 t=cosθ  (1t2)21/t2=t22 t2 (1t2)21=t42t2 t62t4+t21=t42t2 t63t4+3t21=0 (t1)3(t+1)3=0  t1=1 t2=1  x1=arccos(t1)=arccos1=0 x2=arccos(t2)=arccos((1))3.1416 kZ  θ=x=k π=k 3.1416\sin^4 θ - 1/\cos^2 θ=\cos^2 θ - 2 \ \\ (1 - \cos^2 θ)^2 - 1/\cos^2 θ=\cos^2 θ - 2 \ \\ t=\cos θ \ \\ \ \\ (1-t^2)^2 - 1/t^2=t^2 -2 \ \\ t^2 \cdot \ (1-t^2)^2 - 1=t^4 -2t^2 \ \\ t^6-2t^4+t^2 - 1=t^4 -2t^2 \ \\ t^6-3t^4+3t^2 - 1=0 \ \\ (t-1)^3(t+1)^3=0 \ \\ \ \\ t_{1}=1 \ \\ t_{2}=-1 \ \\ \ \\ x_{1}=\arccos (t_{1})=\arccos 1=0 \ \\ x_{2}=\arccos (t_{2})=\arccos ((-1)) \doteq 3.1416 \ \\ k \in Z \ \\ \ \\ θ=x=k \cdot \ \pi=k \cdot \ 3.1416



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