IF a1,a2,a3........,an are in A.p with d as common difference ,then the sum of series 

sind(cosec a1.cosec a2+ cosec a2.cosec a3+......+cosec an-1.cosec an)= cot a1-cot an 

 

Prove with each step properly written and explained ...thanks ....if q. not clear pls refer to rd sharma class 11 arithmetic progression ex22 pg.19.11

Asked by conveyrajat | 15th May, 2014, 08:26: PM

Expert Answer:

Given:
a subscript 1 comma space a subscript 2 space comma space a subscript 3 space...... space a subscript n space b e space A. P space w i t h space d space a s space c o m m o n space d i f f e r e n c e comma space t h e n  d equals a subscript 2 space end subscript minus space a subscript 1 space equals space a subscript 3 space minus space a subscript 2 space space......... space a subscript n minus a subscript n minus 1 end subscript  sin d open parentheses cos e c a subscript 1 cos e c a subscript 2 space plus space cos e c a subscript 2 cos e c a subscript 3 space plus space.... plus space cos e c a subscript n minus 1 end subscript space cos e c a subscript n space close parentheses  equals fraction numerator sin d over denominator sin a subscript 1 sin a subscript 2 end fraction plus fraction numerator sin d over denominator sin a subscript 2 sin a subscript 3 end fraction space plus space..... space plus space fraction numerator sin d over denominator sin a subscript n minus 1 end subscript sin a subscript n end fraction  equals fraction numerator sin open parentheses a subscript 1 minus a subscript 2 close parentheses over denominator sin a subscript 1 sin a subscript 2 end fraction plus fraction numerator sin open parentheses a subscript 2 minus a subscript 3 close parentheses over denominator sin a subscript 2 sin a subscript 3 end fraction space plus space..... space plus space fraction numerator sin open parentheses a subscript n minus a subscript n minus 1 end subscript close parentheses over denominator sin a subscript n minus 1 end subscript sin a subscript n end fraction  equals fraction numerator sin open parentheses a subscript 1 minus a subscript 2 close parentheses over denominator sin a subscript 1 sin a subscript 2 end fraction plus fraction numerator sin open parentheses a subscript 2 minus a subscript 3 close parentheses over denominator sin a subscript 2 sin a subscript 3 end fraction space plus space..... space plus space fraction numerator sin open parentheses a subscript n minus a subscript n minus 1 end subscript close parentheses over denominator sin a subscript n minus 1 end subscript sin a subscript n end fraction  equals fraction numerator sin a subscript 1 cos a subscript 2 minus cos a subscript 1 sin a subscript 2 over denominator sin a subscript 1 sin a subscript 2 end fraction plus fraction numerator sin a subscript 2 cos a subscript 3 minus cos a subscript 2 sin a subscript 3 over denominator sin a subscript 2 sin a subscript 3 end fraction space plus space..... space plus space fraction numerator sin a subscript n cos a subscript n minus 1 end subscript minus cos a subscript n minus 1 end subscript sin a subscript n over denominator sin a subscript n minus 1 end subscript sin a subscript n end fraction  equals c o t a subscript 1 minus c o t a subscript 2 space plus space c o t a subscript 2 minus c o t a subscript 3 space plus space....... space plus space c o t a subscript n minus 1 end subscript minus c o t a subscript n  equals c o t a subscript 1 space minus space c o t a subscript n

Answered by Avinash Soni | 16th May, 2014, 10:09: AM