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Theorem 2:

THEOREM 2

(i)

07_trigonometric_functions-155.gif (where - 1 < x < 1).

07_trigonometric_functions-156.gif (where - 1 < x < 1).

(ii)

07_trigonometric_functions-157.gif

07_trigonometric_functions-158.gif

(iii)

07_trigonometric_functions-159.gif (where |x| > 1).

07_trigonometric_functions-160.gif (where |x| > 1).

PROOF

We prove the first part of (i) and (iii). Since the derivatives exist we may use implicit differentiation.

(i) Let y = arcsin x. Then

x = sin y, -π/2 ≤ y ≤ π/2, dx = cos y dy. From sin2 y + cos2 y = 1 we get

07_trigonometric_functions-161.gif

Since - π/2 ≤ y ≤ π/2, cos y ≥ 0. Then

07_trigonometric_functions-162.gif

Substituting, 07_trigonometric_functions-163.gif

(iii) Let y = arcsec x.

Then

x = sec y, 0 ≤ y ≤ π,

dx = sec y tan y dy.

From tan2 y + 1 = sec2 y we get

tan y = 07_trigonometric_functions-164.gif = 07_trigonometric_functions-165.gif

Since 0 ≤ y ≤ π, tan y and sec 07_trigonometric_functions-166.gif have the same sign.

Therefore 

sec y tan y ≥ 0

and

07_trigonometric_functions-167.gif

07_trigonometric_functions-168.gif

When we turn these formulas for derivatives around we get some surprising new integration formulas.


Last Update: 2006-11-05