If θ1,θ2,θ3............θnare in A.P then sinθ1+sinθ2+............+sinθncosθ1+cosθ2+............+cosθn

sinθ1+sinθ2+........+sinθn=sin(θ1+θn2)sin(nd2)sin(d2)cosθ1+cosθ2+.......+cosθn=cos(θ1+θn2)sin(nd2)sin(d2)

If θ1,θ2,θ3............θnare in A.P then sinθ1+sinθ2+............+sinθncosθ1+cosθ2+............+cosθn

sinθ1+sinθ2+........+sinθn=sin(θ1+θn2)sin(nd2)sin(d2)cosθ1+cosθ2+.......+cosθn=cos(θ1+θn2)sin(nd2)sin(d2)

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If $θ_{1}, θ_{2}, θ_{3} ............ θ_{n}$ are in A.P then $\frac{\sin θ_{1} + \sin θ_{2} + ............ + \sin θ_{n}}{\cos θ_{1} + \cos θ_{2} + ............ + \cos θ_{n}}$

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$\tan (θ_{1} + θ_{n})$

$\tan (\frac{θ_{1} + θ_{n}}{2})$

$\tan (\frac{θ_{1} - θ_{n}}{2})$

answer is C.

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$\sin θ_{1} + \sin θ_{2} + ...... . . + \sin θ_{n} = \frac{\sin (\frac{θ_{1} + θ_{n}}{2}) \sin (\frac{n d}{2})}{\sin (\frac{d}{2})}$

$\cos θ_{1} + \cos θ_{2} + ....... + \cos θ_{n} = \frac{\cos (\frac{θ_{1} + θ_{n}}{2}) \sin (\frac{n d}{2})}{\sin (\frac{d}{2})}$

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If θ1,θ2,θ3............θnare in A.P then sinθ1+sinθ2+............+sinθncosθ1+cosθ2+............+cosθn

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If $θ_{1}, θ_{2}, θ_{3} ............ θ_{n}$ are in A.P then $\frac{\sin θ_{1} + \sin θ_{2} + ............ + \sin θ_{n}}{\cos θ_{1} + \cos θ_{2} + ............ + \cos θ_{n}}$