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The valuie of sin-1 cos(sin-1 x) + cos-1 sin (cos-1 x) is
- a)0
- b)π/4
- c)π/2
- d)π
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The valuie of sin-1 cos(sin-1 x) + cos-1 sin (cos-1 x) isa)0b)π/4c)...
Π/4
c)π/2
d)π
We can use the trigonometric identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to simplify the expression:
sin-1(cos(sin-1 x)) = sin-1(sin(π/2 - sin-1 x))
= sin-1(cos(sin-1 x)) = sin-1(cos(sin-1 x))
= π/2 - sin-1(x)
cos-1(sin(cos-1 x)) = cos-1(cos(π/2 - cos-1 x))
= cos-1(sin(cos-1 x)) = cos-1(sin(cos-1 x))
= π/2 - cos-1(x)
So, the expression simplifies to:
(π/2 - sin-1(x)) * (π/2 - cos-1(x))
Using the trigonometric identity sin(π/2 - θ) = cos(θ) and cos(π/2 - θ) = sin(θ), we can rewrite this as:
(sin-1(x) - π/2) * (cos-1(x) - π/2)
= - (π/2 - sin-1(x)) * (π/2 - cos-1(x))
Therefore, the value of the expression is the negative of the value we found earlier, which is:
- (π/2 - sin-1(x)) * (π/2 - cos-1(x))
= - π/4
So, the answer is (b) π/4.
c)π/2
d)π
We can use the trigonometric identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to simplify the expression:
sin-1(cos(sin-1 x)) = sin-1(sin(π/2 - sin-1 x))
= sin-1(cos(sin-1 x)) = sin-1(cos(sin-1 x))
= π/2 - sin-1(x)
cos-1(sin(cos-1 x)) = cos-1(cos(π/2 - cos-1 x))
= cos-1(sin(cos-1 x)) = cos-1(sin(cos-1 x))
= π/2 - cos-1(x)
So, the expression simplifies to:
(π/2 - sin-1(x)) * (π/2 - cos-1(x))
Using the trigonometric identity sin(π/2 - θ) = cos(θ) and cos(π/2 - θ) = sin(θ), we can rewrite this as:
(sin-1(x) - π/2) * (cos-1(x) - π/2)
= - (π/2 - sin-1(x)) * (π/2 - cos-1(x))
Therefore, the value of the expression is the negative of the value we found earlier, which is:
- (π/2 - sin-1(x)) * (π/2 - cos-1(x))
= - π/4
So, the answer is (b) π/4.
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