Analytic methodsi have to prove that sin^1 x + cos^-1 x = 90 degrees, -1<=x<=1 and my tutor said to use differentiation. How come ?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Yes, the problem could be solved using calulus methods.

For instance, if we'll consider the expression (arcsin x + arccos x) as a function f(x), in order to demonstrate that f(x)=pi/2 is a constant function, we'll have to calculate the first derivative of this function f(x).

If this derivative is cancelling, that means that f(x)=pi/2, is a constant function, knowing the fact that a derivative of a constant function is 0.

We'll take the first derivative of the given function:

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

f'(x) = 1/sqrt(1-x^2) - 1/sqrt(1-x^2)

f'(x)=0 => so f(x)=constant

We notice that for x = 1, the sum of inverse trigonometric functions is pi/2.

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

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