inverse trigonometric functions?

Question

1) if Ʃ( cos^( -1) ai) (i=1 to n) =0 then what is the value of Ʃ( ai) (i=0 to n) ?  [ i is subscript].
2) find range of f(x)= sin^(-1)x + tan^(-1) x + sec^(-1) x.
3) if u= cot^(-1)(√tan a) - tan^(-1)(√tan a) , then tan(π/4 - u/2) is equal to
4) value of x for which sin( cos^(-1) (1+x)) = cos(tan^(-1) x)

any help will be greatly appreciated. Thank you.

Raffaele · Accepted Answer

1)
cos^( -1) ranges from 0 to π
thus is Ʃ( cos^( -1) ai) = 0, ai are 0

2)
domain of f(x) is the intersection of the domains of
sin^(-1)x  ---> [-1,1]
tan^(-1) x ----> R 
sec^(-1) x ----> (-∞,-1]∪[1,+∞)

thus the intersection is just the two values {-1,1}
the range is the set {π/4, 3π/4}

3.

u = cot^(-1)(√tan a) - tan^(-1)(√tan a)
u = (π/2 - tan^(-1)(√tan a)) - tan^(-1)(√tan a)
u = π/2 - 2tan^(-1)(√tan a)
u/2 = π/4 - tan^(-1)(√tan a)
π/4 - u/2 = tan^(-1)(√tan a)
tan (π/4 - u/2) = tan tan^(-1)(√tan a) = √tan a

4.
x = - √(√2/2 - 1/4) - √2/2 - 1/2
x = √(√2/2 - 1/4) - √2/2 - 1/2

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inverse trigonometric functions?

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