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Expand log(sinx) in powers of (x-2) by T...

Expand `log(sinx)` in powers of `(x-2)` by Taylor's theorem.

Text Solution

Verified by Experts

Here, `f(x) = log(sinx)`
`:. f(x-2) = log(sin(x-2))`
So, from Taylor`'`s theorem,
`f(x-2) = f(x) +((-2)/(1!)) f'(x)+((-2)^2/(2!))f''(x)+((-2)^3/(3!))f'''(x)+...`
Here, `f'(x) = (1/sinx)(cosx) = cotx`
`f''(x) = -cosec^2x`
`f'''(x) = 2cosec^2xcotx`
...
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