Prove the following

Question

cosec 2A + cosec 4A + cosec 8A = cot A — cot 8A

Answer 1

This is in my advanced trig book by durell.

general formula for finite series specified

is csc2A+csc4A+...+csc(2ⁿA)=cotA-cot(2ⁿA),,,n>1 or=1

it is derived thusly

csc2A = [2cos²A-cos2A}/2sinAcosA,,,,,the numerator = 1 and the denominator = sin2A

csc2A=cotA-cos2A/2sinAcosA,,,,,,,,change denominator now to sin2A on second term

csc2A=cotA-cot2A

likewise csc4A=cot2A-cot4A

and csc8A=cot4A-cot8A

and on and on, it is a collapsing series meaning the middle terms subtract out

csc2A+csc4A+csc8A=cotA-cot8A

Answer 2

cosec2A = cotA - cot2A

cosec4A = cot2A - cot4A

cosec8A = cot4A - cot8A

Adding this 3 identities we get:

cosec2A + cosec4A + cosec8A = cotA - cot8A.

!st three identities easy prove expressing all in terms of sin and cos