Question

differentiation of 1 dx /1 - cos x
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Answer 1

we have to differentiate $f(x)=\frac{1}{1-cosx}$ right ?

can we write f(x) = 1/(1 - cosx) = (1- cosx)^-1

now, using concepts : if a function y = f(x)^n is given then, differentiation of y = dy/dx = n . f(x)^(n - 1) × df(x)/dx

and also you should know differentiation of cosx = -sinx

now, differentiating f(x) = 1/(1 - cosx) or (1 - cosx)^-1

df(x)/dx = d{(1 - cosx)^-1}/dx

= -1(1 - cosx)^{-1 - 1} × d(1 - cosx)/dx

= -1(1 - cosx)^-2 × (0 -(-sinx))

= -1/(1 - cosx)² × sinx

= -sinx/(1 - cosx)²

hence, differentiation of 1/(1 - cosx) = -sinx/(1 - cosx)²