Source: MAT $2011$
I cannot decide between two of the MCQ options. Both of them seem correct.
(a) $\cos\alpha = \sin(\beta+\gamma)$
(b) $\sin\beta = \sin\alpha \sin\gamma$
(a) If I join the centre of the circle to the point where the tangent meets the circle, I'll get a right angled triangle. Thus, angle ACB is $90 + \alpha$ degrees. Now $$\alpha + \beta + \gamma = 90$$ $$or, \sin(\beta+\gamma) = \cos\alpha$$
(b) WLOG, let radius of circle = $1$. Then $$AC = \frac{1}{\sin\alpha}$$ Applying sine rule in $\triangle ABC$ $$\sin\beta = \frac{\sin\gamma}{\frac{1}{\sin\alpha}}$$
The answer key says the answer is (b). Then what am I doing wrong in (a)?