prove that 1/sin1sin2 + 1/sin2sin3 +......+1/sin89sin90 = cos1/sin^21
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Solution:
LHS
Now,
Using the trigonometric identity
sin (a+b)= sin a cos b +cos a sin b
sin(a-b)=sin a cos b - cos a sin b
Similarly, other terms can be written as
cot 2-cot 3, cot 3-cot 4,........... , cot 89-cot 90.
So, adding all these terms , we get
→→cot 1 - cot 2 +cot 2-cot 3 +cot 3 - cot 4 +cot 4 - cot 5 +........+cot 88 - cot 89+cot 89 -cot 90
Cancelling out the like terms
→cot 1 - cot 90
As, sin 1 and sin 90 Radian, are approximately same, so you can write it as
= R H S
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