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検算

作成: 2020-09-30 更新: 2020-09-30

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検算 \[ ( \tau_s ( y^2 + z^2 ) + \tau_c ( x y ) )^2 \\ + ( - \tau_s ( x y ) - \tau_c ( z^2 + x^2 ) )^2 \\ + ( - \tau_s ( z x ) + \tau_c ( y z ) )^2 \\ \ \\ \ \\ = (\tau_s)^2 ( y^2 + z^2 )^2 + 2 \tau_s ( y^2 + z^2 ) \tau_c ( x y ) + (\tau_c)^2 x^2 y^2 \\ + (\tau_s)^2 x^2 y^2 +2 \tau_s ( x y ) \tau_c ( z^2 + x^2 ) + (\tau_c)^2 ( z^2 + x^2 )^2 \\ + ( \tau_s)^2 z^2 x^2 -2 \tau_s ( z x ) \tau_c ( y z ) + (\tau_c)^2 y^2 z^2 \\ \ \\ \ \\ = (\tau_s)^2 ( y^4 + 2 y^2 z^2 + z^4 ) + 2 \tau_s \tau_c x y^3 + 2 \tau_s \tau_c x y z^2 + (\tau_c)^2 x^2 y^2 \\ + (\tau_s)^2 x^2 y^2 + 2 \tau_s \tau_c x y z^2 + 2 \tau_s \tau_c x^3 y + (\tau_c)^2 ( z^4 + 2 z^2 x^2 + x^4 ) \\ + (\tau_s)^2 z^2 x^2 - 2 \tau_s \tau_c x y z^2 + (\tau_c)^2 y^2 z^2 \\ \ \\ \ \\ = (\tau_s)^2 y^4 + 2 (\tau_s)^2 y^2 z^2 + (\tau_s)^2 z^4 \\ + 2 \tau_s \tau_c x y^3 + 2 \tau_s \tau_c x y z^2 \\ + (\tau_c)^2 x^2 y^2 \\ \ \\ + (\tau_s)^2 x^2 y^2 + 2 \tau_s \tau_c x y z^2 + 2 \tau_s \tau_c x^3 y \\ + (\tau_c)^2 z^4 + 2 (\tau_c)^2 z^2 x^2 + (\tau_c)^2 x^4 \\ \\ \ \\ + (\tau_s)^2 z^2 x^2 - 2 \tau_s \tau_c x y z^2 + (\tau_c)^2 y^2 z^2 \\ \ \\ \ \\ = (\tau_c)^2 x^4 + (\tau_s)^2 y^4 + (\tau_c)^2 z^4 + (\tau_s)^2 z^4 \\ \ \\ + 2 \tau_s \tau_c x^3 y \\ + 2 \tau_s \tau_c x y^3 \\ \ \\ + (\tau_s)^2 x^2 y^2 + (\tau_c)^2 x^2 y^2 + 2 (\tau_s)^2 y^2 z^2 + (\tau_c)^2 y^2 z^2 + (\tau_s)^2 z^2 x^2 + 2 (\tau_c)^2 z^2 x^2 \\ \ \\ + 2 \tau_s \tau_c x y z^2 \\ + 2 \tau_s \tau_c x y z^2 - 2 \tau_s \tau_c x y z^2 \\ \ \\ \ \\ = (\tau_c)^2 x^2 ( 1 - y^2 - z^2 ) + (\tau_s)^2 y^2 ( 1 - z^2 - x^2 ) + z^4 \\ \ \\ + 2 \tau_s \tau_c x y ( x^2 + y^2 ) \\ \ \\ + x^2 y^2 + (\tau_s)^2 y^2 z^2 + y^2 z^2 + (\tau_c)^2 z^2 x^2 + z^2 x^2 \\ \ \\ + 2 \tau_s \tau_c x y z^2 \\ \\ \ \\ \ \\ = (\tau_c)^2 x^2 - (\tau_c)^2 x^2 y^2 - (\tau_c)^2 z^2 x^2 + (\tau_s)^2 y^2 - (\tau_s)^2 y^2 z^2 - (\tau_s)^2 y^2 x^2 + z^2 ( 1 - x^2 - y^2 ) \\ \ \\ + x^2 y^2 + (\tau_s)^2 y^2 z^2 + y^2 z^2 + (\tau_c)^2 z^2 x^2 + z^2 x^2 + 2 \tau_s \tau_c x y \\ \ \\ = (\tau_c)^2 x^2 + (\tau_s)^2 y^2 + z^2 + 2 \tau_s \tau_c x y \\ \ \\ = (\tau_c)^2 x^2 + (\tau_s)^2 y^2 + 1 - x^2 - y^ 2 + 2 \tau_s \tau_c x y \\ \ \\ = 1 - ( 1- (\tau_c)^2 ) x^2 ) - ( 1 - (\tau_s)^2 ) y^2 + 2 \tau_s \tau_c x y \\ \ \\ = 1 - (\tau_s)^2 x^2 ) - (\tau_c)^2 y^2 + 2 \tau_s \tau_c x y \\ \ \\ = 1 - ( \tau_s x - \tau_c y )^2 \\ \]