Question

Using only what you know about the convergent power series expansions of the sine and cosine functions prove the following identity:
Cos^2(x)+Sin^2(x)

Answer 1

\[\large \cos x = 1-\frac{ x^2 }{ 2! }+\frac{ x^4 }{ 4! }-\frac{ x^6 }{ 6! }-\frac{ x^8 }{ 8! }-...\] \[\cos^2x=1+\left[ 2\left( \frac{ -1 }{ 2! } \right) \right]x^2+\left[ 2\left( \frac{ 1 }{ 4! } \right) +\left( \frac{ -1 }{ 2! } \right)^2\right]x^4+2\left[ \left( \frac{ -1 }{ 6! } \right) + \left( \frac{ -1 }{ 2! } \right) \left( \frac{ 1 }{ 4! } \right)\right] x^6+...\]

Answer 2

I'm having trouble seeing what you wrote. It goes off to the left. off the screen. o-O

Answer 3

\[\large \sin x = x-\frac{ x^3 }{ 3! }+\frac{ x^5 }{ 5! }-\frac{ x^7 }{ 7! }+...\] \[\sin^2x=x^2+\left[ 2\left( \frac{ -1 }{ 3! } \right) \right]x^4+\left[ 2\left( \frac{ 1 }{ 5! } \right) +\left( \frac{ -1 }{ 3! } \right)^2 \right]x^6+...\]

Answer 4

that's better

Answer 5

you can do the expansion of (cos x)^2 and (sin x)^2 on paper. check if the sum of the coefficients of x^2, x^4, x^6, and so on, equals 0.

Answer 6

Ahhhh. I see. Thank you very much!

Answer 7

thanks