First slide

Trigonometric transformations

Question

If x, y, z are in A.P, then $\frac{\sin x - \sin z}{\cos z - \cos x}$ is equal to

Moderate

A

tan y
B

cot y
C

sin y
D

cos y

Solution

$\begin{aligned} \frac{\sin x - \sin z}{\cos z - \cos x} = \frac{2 \cos (\frac{x + z}{2}) \sin (\frac{x - z}{z})}{2 \sin (\frac{x + z}{z}) \sin (\frac{x - z}{z})} \\ = \cot (\frac{x + z}{2}) = \cot (y) \end{aligned}$

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