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Show that sin (cot^(-1) (tan cos^(-1)x))...

Show that `sin (cot^(-1) (tan cos^(-1)x)) =x`

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For `theta = cos^(-1) x`, we have the following triangle

`:. E = sin (cot^(-1) (tan (cos^(-1) x)))`
`= sin (cot^(-1) (tan (tan^(-1).(sqrt(1 - x^(2)))/(x))))`
`= sin (cot^(-1).(sqrt(1 - x^(2)))/(x))`
Now, for `alpha = cot^(-1) (sqrt(1 - x^(2)))/(x)`, we have following triangle

`:. E = sin (sin^(-1).(x)/(1)) = x`
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CENGAGE PUBLICATION-INVERSE TRIGONOMETRIC FUNCTIONS-All Questions
  1. Write tan^(-1) x, x gt 0 in the form of other inverse trigonometric fu...

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  2. Find tan^(-1). (x)/(sqrt(a^(2) - x^(2))) in terms of sin^(-1), where x...

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  3. Show that sin (cot^(-1) (tan cos^(-1)x)) =x

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  4. If x<0, then prove that cos^(-1)x=pi-sin^(-1)sqrt(1-x^2)

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  5. Prove that cos^(-1) {sqrt((1 + x)/(2))} = (cos^(-1) x)/(2) , -1 lt x l...

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  6. Prove that tan^(-1){x/(a+sqrt(a^2-x^2))}=1/2sin^(-1)(x/a) ,-a lt x lt ...

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  7. Prove that: sin^(-1){(sqrt(1+x)+sqrt(1-x))/2}=pi/4+(cos^(-1)x)/2,""0 <...

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  8. Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2tan^(-1)x^n ,0 lt x llt oo

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  9. If x in [-1,0], then find the value of cos^(-1)(2x^2-1)-2sin^(-1)x

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  10. If 1/(sqrt(2)) < x < 1, then prove that cos^(-1)x+cos^(-1)((x+sqrt(1-x...

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  11. Find the value of sin^(-1)(sin5)+cos^(-1)(cos10)+tan^(-1){tan(-6)}+cot...

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  12. Find the minimum value of the function f(x)=(pi^2)/(16cot^(-1)(-x))-co...

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  13. Find the range of y=(cot^(-1)x)(cot^(-1)(-x))dot

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  14. The value of 2tan^(-1)(cos e ctan^(-1)x-tancot^(-1)x) is equal to cot^...

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  15. Prove that costan^(-1)sincot^(-1)x=sqrt((x^2+1)/(x^2+2))

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  16. If sin^(-1)x=pi/5, for some x in (-1,1), then find the value of cos^(-...

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  17. If sin(sin^(-1)(1/5)+cos^(-1)x)=1, then find the value of x.

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  18. Solve sin^(-1)xlt=cos^(-1)xdot

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  19. Find the range of f(x)=sin^(-1)x+tan^(-1)x+cos^(-1)xdot

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  20. Find the minimum value of (sec^(-1)x)^2+("c o s e c"^(-1))^2

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