If α, −π/2 < α < π/2 is the solution of 4cosθ + 5sinθ = 1, then the value of tanα is

Question

If \(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\) is the solution of \(4 \cos \theta+5 \sin \theta=1\), then the value of \(\tan \alpha\) is

(1) \(\frac{10-\sqrt{10}}{6}\)

(2) \(\frac{10-\sqrt{10}}{12}\)

(3) \(\frac{\sqrt{10}-10}{12}\)

(4) \(\frac{\sqrt{10}-10}{6}\)

Answer 1

Correct option is (3) \(\frac{\sqrt{10}-10}{12}\)

\(4+5 \tan \theta=\sec \theta\)

Squaring: \(24 \tan ^{2} \theta+40 \tan \theta+15=0\)

\(\tan \theta=\frac{-10 \pm \sqrt{10}}{12}\)

and \(\tan \theta=-\left(\frac{10+\sqrt{10}}{12}\right)\) is Rejected.