Specimen Cyprus Seniors Provincial/2nd grade/Problem 2

Problem

If $\alpha=\sin x_{1}$ , $\beta=\cos x_{1}\sin x_{2}$ , $\gamma=\cos x_{1}\cos x_{2} \sin x_{3}$ and $\delta=\cos x_{1}\cos x_{2}\cos x_{3}$ prove that $\alpha^2+\beta^2+\gamma^2+\delta^2=1$

Solution

Plug in the expressions for $\alpha$ , $\beta$ , $\gamma$ , and $\delta$ .

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + \cos^2x_{1}\cos^2x_{2} \sin^2x_{3} + \cos^2x_{1}\cos^2x_{2} \cos^2x_{3}$

Factor the last two terms:

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + (\cos^2x_{1}\cos^2x_{2})(\sin^2x_{3}+\cos^2x_{3})$

Use the identity $\cos^2x + \sin^2x = 1$ :

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $\sin^2x_1 + \cos^2x_{1}\sin^2x_{2} + \cos^2x_{1}\cos^2x_{2}$

Factor the last two terms:

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $\sin^2x_1 + (\cos^2x_{1})(\sin^2x_{2}+\cos^2x_{2})$

Use the identity $\cos^2x + \sin^2x = 1$ :

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $\sin^2x_1 + \cos^2x_{1}$

Use the identity $\cos^2x + \sin^2x = 1$ :

$\alpha^2+\beta^2+\gamma^2+\delta^2=$ $1$

$\boxed{\mathbb{Q.E.D}}$

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Specimen Cyprus Seniors Provincial/2nd grade/Problem 2

Problem

Solution

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