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Prove that cos^(-1) ((1 - x^(2n))/(1 + x...

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

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To prove that \[ \cos^{-1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right) = 2 \tan^{-1}(x^n), \quad 0 < x < \infty, \] we will follow these steps: ...
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CENGAGE ENGLISH-INVERSE TRIGONOMETRIC FUNCTIONS-Illustration
  1. Prove that tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1)...

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  2. Prove that sin^(-1) {(sqrt(1 + x) + sqrt(1 - x))/(2)} = (pi)/(4) + (co...

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  3. Prove that cos^(-1) ((1 - x^(2n))/(1 + x^(2n))) = 2 tan^(-1) x^(n), 0 ...

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

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  5. If (1)/(sqrt2) lt x lt 1, then prove that cos^(-1) x + cos^(-1) ((x + ...

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  6. Find the value of sin^(-1) (sin 5) + cos^(-1) (cos 10) + tan^(-1) {tan...

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

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

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  9. The value of 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) is equal t...

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  10. Prove that cos (tan^(-1) (sin (cot^(-1) x))) = sqrt((x^(2) + 1)/(x^(2)...

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

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

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  13. solve sin^(-1) x le cos^(-1) x

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

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  15. Find the minimum value of (sec^(-1) x)^(2) + (cosec^(-1) x)^(2)

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  16. Find the range of f(x) = (sin^(-1) x)^(2) + 2pi cos^(-1) x + pi^(2)

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  17. Solve sin^(-1). (14)/(|x|) + sin^(-1).(2 sqrt15)/(|x|) = (pi)/(2)

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  18. If alpha = sin^(-1) (cos(sin^-1) x)) and beta = cos^(-1) (sin (cos^(-1...

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  19. If sec^(-1) x = cosec^(-1) y, then find the value of cos^(-1).(1)/(x) ...

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  20. Prove that tan^(-1) x + tan^(-1).(1)/(x) = {(pi//2,"if" x gt 0),(-pi//...

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