arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))

We will learn how to prove the property of the inverse trigonometric function arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))


Let, cos\(^{-1}\) x = α and cos\(^{-1}\) y = β

From cos\(^{-1}\) x = α we get,

x = cos α

and from cos\(^{-1}\) 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 + \(\sqrt{1 - cos^{2} α}\) \(\sqrt{1 - cos^{2} β}\)

⇒ cos (α - β) = (xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))

⇒ α - β = cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))

or, cos\(^{-1}\) x - cos\(^{-1}\) y = cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))

Therefore, arccos(x) - arccos(y) = arccos(xy) + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))       Proved.


Note: If x > 0, y > 0 and x\(^{2}\) + y\(^{2}\) > 1, then the cos\(^{-1}\) x + sin\(^{-1}\) y may be an angle more than π/2 while cos\(^{-1}\)(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), is an angle between – π/2 and π/2.

Therefore, cos\(^{-1}\) x - cos\(^{-1}\) y = π - cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))

 Inverse Trigonometric Functions

11 and 12 Grade Math

From arccos(x) - arccos(y) to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. What is a Triangle? | Types of Triangle | Scalene Triangle | Isosceles

    Jun 17, 24 11:22 PM

    What is a triangle
    A simple closed curve or a polygon formed by three line-segments (sides) is called a triangle. The above shown shapes are triangles. The symbol of a triangle is ∆. A triangle is a polygon with three s…

    Read More

  2. Interior and Exterior of an Angle | Interior Angle | Exterior Angle

    Jun 16, 24 05:20 PM

    Interior of an Angle
    Interior and exterior of an angle is explained here. The shaded portion between the arms BA and BC of the angle ABC can be extended indefinitely.

    Read More

  3. Angles | Magnitude of an Angle | Measure of an angle | Working Rules

    Jun 16, 24 04:12 PM

    Naming an Angle
    Angles are very important in our daily life so it’s very necessary to understand about angle. Two rays meeting at a common endpoint form an angle. In the adjoining figure, two rays AB and BC are calle

    Read More

  4. What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

    Jun 16, 24 02:34 PM

    Square - Polygon
    What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.

    Read More

  5. Simple Closed Curves | Types of Closed Curves | Collection of Curves

    Jun 16, 24 12:31 PM

    Closed Curves Examples
    In simple closed curves the shapes are closed by line-segments or by a curved line. Triangle, quadrilateral, circle, etc., are examples of closed curves.

    Read More