First slide

Compound angles

Question

If $\sin θ_{1} \sin θ_{2} - \cos θ_{1} \cos θ_{2} + 1 = 0$ then the value of $\tan (θ_{1} / 2) \cot (θ_{2} / 2)$ is equal to

Moderate

A

-1
B

1
C

2
D

-2

Solution

$\begin{array}{l} \sin θ_{1} \sin θ_{2} - \cos θ_{1} \cos θ_{2} = - 1 \\ or \cos (θ_{1} + θ_{2}) = 1 \\ \Rightarrow θ_{1} + θ_{2} = 2 nπ, n \in 1 \\ or \frac{θ_{1}}{2} + \frac{θ_{2}}{2} = nπ \end{array} Thus, \tan \frac{θ_{1}}{2} \cot \frac{θ_{2}}{2} = \tan \frac{θ_{1}}{2} \cot (nπ - \frac{θ_{1}}{2}) = - \tan \frac{θ_{1}}{2} \cot \frac{θ_{1}}{2} = - 1$

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