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Chapter 2: Problem 137

If \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi\) prove that \(x^{2}+y^{2}+z^{2}+2 x y z=1\)

Short Answer

Expert verified
Given the equation \(\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi\), the equation \(x^{2}+y^{2}+z^{2}+2xyz=1\) can be derived using the properties of the trigonometric function cosine and the transformation of the equation by squaring.

Step by step solution

01

Rearrange the equation

Given the equation \(\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi\), we rearrange it into two simple expressions by using the property of cosine: \(\cos(\pi - x)= - \cos(x)\): \(\cos^{-1}x + \cos^{-1}y = \pi - \cos^{-1}z\). Applying cosine to both sides, we obtain: \(cos(\cos^{-1}x + \cos^{-1}y) =cos(\pi - \cos^{-1}z)\). By using the cosine of sum formula where \(\cos(A+B)= \cos(A)\cos(B) - \sin(A)\sin(B)\), the left hand side of the equation becomes \(x \cdot y - \sqrt{(1-x^{2})} \cdot \sqrt{(1-y^{2})}\)
02

Apply the cosine operation on the left side

Applying the cosine operation on the right hand side of the equation, it becomes \(cos(\pi - \cos^{-1}z)= -z\). So, the equation becomes \(x \cdot y - \sqrt{(1-x^{2})} \cdot \sqrt{(1-y^{2})}= -z\)
03

Square both sides of the equation

This equation has to be rearranged and then both sides need to be squared to eliminate the radical expressions. This results in: \((x^{2} \cdot y^{2} - 2 \cdot x \cdot y \cdot z + z^{2}) = (1- x^{2}) \cdot (1 - y^{2})\)
04

Simplify the final equation

The equation is simplified by expanding the brackets on the right side and collecting like terms, resulting in: \((x^{2}y^{2}+z^{2}+ 2xyz)= (1- x^{2} - y^{2}+ x^{2}y^{2})\). Upon further simplification: \(x^{2}+y^{2}+z^{2}+2xyz\)=1

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is a fundamental part of trigonometry, often dealing with the measurement of angles and the length of the adjacent side to a hypotenuse in a right triangle. In this exercise, we are dealing with the inverse cosine, denoted as \( \cos^{-1}(x) \), which gives the angle whose cosine is \((x)\). This comes into play when examining angles that sum up to \( \pi \).

This property is derived from how cosine operates over different quadrants on the unit circle:
  • Angles in the first and second quadrants have positive and negative cosine values, respectively.
  • The identity \( \cos(\pi - x) = -\cos(x) \) helps transition between these quadrants effectively.
These characteristics of the cosine function are crucial when verifying trigonometric identities or performing algebraic manipulation within trigonometric proofs.
Trigonometric Identities
Trigonometric identities are equations that hold true for any angle and are used to simplify complex trigonometric equations. One such identity is the cosine addition formula: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \). This formula is vital in the provided solution where it helps combine angles: \( \cos^{-1}x + \cos^{-1}y \).

These identities allow us to transform expressions to a more manageable form:
  • The sum of angles in this problem is transformed with the help of known cosine properties.
  • The identity \( \cos(\pi - \theta) = -\cos(\theta) \) is used to express certain angles in terms of cosine.
Understanding and applying these identities can help prove complex trigonometric equations succinctly and efficiently.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to reveal insights or solve for unknown variables. In the solution to the exercise, algebraic manipulation is a key technique. Here鈥檚 how it helps:
  • First, the solution uses trig identities to convert the expressions into algebraic forms.
  • Then, we rearrange and square both sides of the initial trigonometric equation, enabling the elimination of square roots, simplifying the process.
  • By expanding and combining like terms in the equation, necessary simplifications lead us to the desired identity: \( x^2 + y^2 + z^2 + 2xyz = 1 \).
Through careful steps in algebraic manipulation, we can handle complex problems incrementally, making higher-level mathematics like trigonometry accessible and understandable.

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Most popular questions from this chapter

Prove that: If \(\tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\frac{\pi}{2}\), then prove that \(x y+\) \(y z+z x=1\).

Prove that \(\sin \left(\frac{1}{2} \cos ^{-1}\left(\frac{1}{9}\right)\right)=\frac{2}{3}\)

Find the value of $$ \sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)-\sin ^{-1}\left(\frac{63}{65}\right) $$

Prove that: $$ \cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right)=\frac{\pi}{3}, \frac{1}{2}

Prove that: If \(\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}\), then prove that \(x+y+x y=1\).
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