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arccos(x) - arccos(y) = arccos(xy + 1x21y2)

We will learn how to prove the property of the inverse trigonometric function arccos(x) - arccos(y) = arccos(xy + 1x21y2)


Proof:  

Let, cos1 x = α and cos1 y = β

From cos1 x = α we get,

x = cos α

and from cos1 y = β we get,

y = cos β

Now, cos (α - β) = cos α cos β + sin α sin β

⇒ cos (α - β) = cos α cos β + 1cos2α 1cos2β

⇒ cos (α - β) = (xy + 1x21y2)

⇒ α - β = cos1(xy + 1x21y2)

or, cos1 x - cos1 y = cos1(xy + 1x21y2)

Therefore, arccos(x) - arccos(y) = arccos(xy) + 1x21y2)       Proved.

 

Note: If x > 0, y > 0 and x2 + y2 > 1, then the cos1 x + sin1 y may be an angle more than π/2 while cos1(xy - 1x21y2), is an angle between – π/2 and π/2.

Therefore, cos1 x - cos1 y = π - cos1(xy + 1x21y2)

 Inverse Trigonometric Functions






11 and 12 Grade Math

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