If a1- a2- a3- - an are in A-P- with common difference d- then the sum of the series sin d-cosec a1cosec a2-cosec a1cosec a3- -cosec an-1- cosec-an- isa a1- anb cosec a1-cosec anc a1- and tan a1-tan an

(c) cot a1 minus; cot an We have: sin #160;d #160;cosec #160;a1 #160;cosec #160;a2+cosec #160;a2 #160;cosec #160;a3+....+cosec #160;an 1 ...

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Byju's Answer

Standard IX

Mathematics

Calculating Heights and Distances

If a1, a2, a3...

Question

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [cosec a₁ cosec a₂ + cosec a₁ cosec a₃ + .... + cosec a_n_{− 1} cosec a_n] is
(a) sec a₁ − sec a_n
(b) cosec a₁ − cosec a_n
(c) cot a₁ − cot a_n
(d) tan a₁ − tan a_n

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Solution

(c) cot a₁ − cot a_{n

We have:

$\sin d (\cos e c a_{1} \cos e c a_{2} + \cos e c a_{2} \cos e c a_{3} + . . . . + \cos e c a_{n - 1} \cos e c a_{n}) = \frac{\sin d}{\sin a_{1} \sin a_{2}} + \frac{\sin d}{\sin a_{2} \sin a_{3}} + . . . . . + \frac{\sin d}{\sin a_{n - 1} \sin a_{n}} = \frac{\sin (a_{2} - a_{1})}{\sin a_{1} \sin a_{2}} + \frac{\sin (a_{3} - a_{2})}{\sin a_{2} \sin a_{3}} + . . . . + \frac{\sin (a_{n} - a_{n - 1})}{\sin a_{n - 1} \sin a_{n}} = \frac{\sin a_{2} \cos a_{1} - \cos a_{2} \sin a_{1}}{\sin a_{1} \sin a_{2}} + \frac{\sin a_{3} \cos a_{2} - \cos a_{3} \sin a_{2}}{\sin a_{1} \sin a_{2}} + . . . . . + \frac{\sin a_{2} \cos a_{1} - \cos a_{2} \sin a_{1}}{\sin a_{1} \sin a_{2}} = (\cot a_{1} - \cot a_{2}) + (\cot a_{2} - \cot a_{3}) + . . . . . + (\cot a_{n - 1} - \cot a_{n}) = \cot a_{1} - \cot a_{n}$}

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If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1 cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is (a) sec a1 − sec an (b) cosec a1 − cosec an (c) cot a1 − cot an (d) tan a1 − tan an

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [cosec a₁ cosec a₂ + cosec a₁ cosec a₃ + .... + cosec a_n_{− 1} cosec a_n] is
(a) sec a₁ − sec a_n
(b) cosec a₁ − cosec a_n
(c) cot a₁ − cot a_n
(d) tan a₁ − tan a_n