HELP ME, PLEASE. I REALLY NEED IT Find x: S=(1)/(sin2x)+(1)/(sin4x)+(1)/(sin8x)+...+(1)/(sin2^(n)x )

Question

HELP ME, PLEASE. I REALLY NEED IT

Find x:

`S=(1)/(sin2x)+(1)/(sin4x)+(1)/(sin8x)+...+(1)/(sin2^(n)x )`

Answer 1

• cotx - cot2ⁿx

Explanation:

• Note: finding the sum not x

First rewrite the first term as:

• 1/sin2x =
• sinx / (sinxsin2x) =
• sin(2x-x) / (sinxsin2x) =
• (sin2xcosx - cosxsinx) / (sinxsin2x) =
• (sin2xcosx)/(sinxsin2x) - (cos2xsinx)/(sinxsin2x) =
• cosx/sinx - cos2x/sin2x =
• cotx - cot2x

Now the sum is:

• S = cotx - cot2x + cot2x - cot4x +... + cot2ⁿ⁻²x - cot2ⁿx =
• cotx - cot2ⁿx