Solve the problem that involves ##\cos^{-1} x + \cos^{-1}y##

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The discussion focuses on solving the equation involving the sum of inverse cosines, specifically ##\cos^{-1} x + \cos^{-1} y##. A right-angled triangle approach is utilized, defining angles C and A corresponding to x and y, respectively, leading to the relationship A + C = π/2. The key derivation shows that the expression simplifies to ##\cos^{-1}(0) = π/2##, indicating that the left-hand side equals the right-hand side. The conversation highlights that the assumption of x and y being on the unit circle is unnecessary for the solution. The final conclusion reaffirms the trigonometric identity used in the calculations.
chwala
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Homework Statement
See attached.
Relevant Equations
Trigonometry
1694237381792.png
In my approach (using a right angled triangle) i let,

##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##

and

##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##

Also, ##A+C = \dfrac{π}{2}##

and ##\cos \dfrac{π}{2}= 0##

##xy - \sqrt{(y^2) ⋅(x^2)}=xy-xy=0##

It follows that,

##\cos^{-1} [xy - \sqrt{(1-x^2)(1-y^2)}]= \cos^{-1}[ xy - \sqrt{(y^2) ⋅(x^2)}]##

##=\cos^{-1} (xy - \sqrt{y^2}⋅ \sqrt{x^2})=\cos^{-1} (xy-xy)=\cos^{-1} (0)= \dfrac{π}{2}##
 
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Cos of the LHS =
\cos(\cos^{-1}x+\cos^{-1}y)=\cos(\cos^{-1}x)\cos(\cos^{-1}y)-\sin(\cos^{-1}x)\sin(\cos^{-1}y)
=xy-\sqrt{1-x^2}\sqrt{1-y^2}
 
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chwala said:
Homework Statement: See attached.
Relevant Equations: Trigonometry

View attachment 331694In my approach (using a right angled triangle) i let,

##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##

and

##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##

Also, ##A+C = \dfrac{π}{2}##
You seem to be assuming that ##\displaystyle x^2+y^2=1 \, , \ ## as if ##(x,\, y) \ ## is an ordered pair on the unit circle. That assumption is not necessary.

If ## \displaystyle \cos^{-1} x = C \, , \ ## then ##\displaystyle \cos C = x \ ## and ##\displaystyle \sin C = \sqrt{ 1-x^2} \ . \ ##

etc.

See Post #2 by @anuttarasammyak .
 
Last edited:
anuttarasammyak said:
Cos of the LHS =
\cos(\cos^{-1}x+\cos^{-1}y)=\cos(\cos^{-1}x)\cos(\cos^{-1}y)-\sin(\cos^{-1}x)\sin(\cos^{-1}y)
=xy-\sqrt{1-x^2}\sqrt{1-y^2}
ok i see the logic now

##\cos(A+B)=\cos A \cos B - \sin A \sin B##.

Cheers Man!
 
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