tan θ (cos θ + cot θ) = sin θ + 1

Question

should use the pythagorean identity

Answer 1

Since tanθ=sinθ/cosθ and cotθ=cosθ/sinθ, sinθ+cos²θ/sinθ=sinθ+1.

The sinθ terms cancel. cos²θ/sinθ=1, 1-sin²θ=sinθ, sin²θ+sinθ-1=0, (sin²θ+sinθ+¼)-¼-1=0, (sinθ+½)²=5/4.

sinθ+½=±√5/2, sinθ=-½±√5/2, sinθ=0.6180 or -1.6180 approx. Note that cosθ=tanθ.

But -1.6180 is out of range for sine, so θ=arcsin(0.6180)=38.17° approx, also 141.83°, and ±360° these angles. However, we must check out each value in the original equation:

θ=38.17: 0.7862(0.7862+1.2720)=1.6180 is correct.

θ=141.83: -0.7862(-0.7862-1.2720)=1.6180 is also correct.