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Sin inverse X + Cot inverse X + tan inverse X Range=????

Sin inverse X + Cot inverse X + tan inverse X Range=????
 

Grade:12

2 Answers

Aditya Gupta
2081 Points
4 years ago
we know the standard identity:
Cot inverse x + tan inverse x= pi/2
also, Sin inverse x lies in [– pi/2, pi/2].
so that range of Sin inverse X + Cot inverse X + tan inverse X= Sin inverse X + pi/2 is [0, pi].
KINDLY APPROVE :))
Vikas TU
14149 Points
3 years ago
f(x)=sin−1x+cos−1x+tan−1x
Using the Inverse Trigonometric Identity,
sin−1x+cos−1x=π/2
We get,
f(x)=π/2+tan−1x
Now,, range of tan−1x as [−π/2,π/2]…
-π/2
so,-π/2+π/2
here it comes , 0
but here it comes a great mistake because
Now most of us apply the range of tan−1x as [−π/2,π/2]
So we get the range of f(x) as [0,π] but there is a very big mistake here.
We know that the domain of f(x)f(x) is [−1,1] hence the term tan−1x wouldn’t hold the values −π/2 and π/2
So the new range becomes,
f(−1)=π/2−π/4=π/4 f(−1)=π2−π/4=π/4
f(1)=π2+π/4=3π/4 f(1)=π/2+π/4=3π/4
So we get f(x)∈[π/4,3π/4]

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