View source for Specimen Cyprus Seniors Provincial/2nd grade/Problem 2

== Problem ==
If <math>\alpha=\sin x_{1}</math>,<math>\beta=\cos x_{1}\sin x_{2}</math>, <math>\gamma=\cos x_{1}\cos x_{2} \sin x_{3}</math> and <math>\delta=\cos x_{1}\cos x_{2}\cos x_{3}</math> prove that <math>\alpha^2+\beta^2+\gamma^2+\delta^2=1</math>


== Solution ==
Plug in the expressions for <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math>, and <math>\delta</math>.

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + \cos^2x_{1}\cos^2x_{2} \sin^2x_{3} + \cos^2x_{1}\cos^2x_{2} \cos^2x_{3}</math> 


Factor the last two terms: 

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + (\cos^2x_{1}\cos^2x_{2})(\sin^2x_{3}+\cos^2x_{3})</math> 


Use the identity <math>\cos^2x + \sin^2x = 1</math>: 

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + \cos^2x_{1}\cos^2x_{2}</math> 


Factor the last two terms: 

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>\sin^2x_1 + (\cos^2x_{1})(\sin^2x_{2}+\cos^2x_{2})</math> 


Use the identity <math>\cos^2x + \sin^2x = 1</math>: 

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>\sin^2x_1 + \cos^2x_{1}</math> 


Use the identity <math>\cos^2x + \sin^2x = 1</math>: 

<math>\alpha^2+\beta^2+\gamma^2+\delta^2=</math> <math>1</math> 

<math>\boxed{\mathbb{Q.E.D}}</math>


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== See also ==
* [[Cyprus Seniors Provincial competition]]
* [[Cyprus Seniors Provincial Problems and Solutions]]
* [[Mathematics competition resources]]