Use Derivative to Show That arcsin(x) + arccos(x) = pi/2

Differentiation is used to prove that arcsin(x) + arccos(x) = pi/2.

Let f(x) = arcsin(x) + arccos(x)

We first prove that f(x) is a constant function. First find the
derivative of f.

f'(x) = d( arcsin(x) )/dx + d( arccos(x) )/dx

= 1 / sqrt(1 - x
2) + (-1 / sqrt(1 - x2) )

= 0

Now if f'(x) = 0 for all values of x, then that means that f(x) is a constant function that may be calculated using any value of x. Let us use x = 0 and x = 1.(Note that one value is enough).

f(0) = arcsin(0) + arccos(0) = 0 + pi/2 = pi/2

f(1) = arcsin(1) + arccos(1) = pi/2 + 0 = pi/2

Hence arcsin(x) + arccos(x) = pi/2 for all values of x.



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