\u5b9f\u6570 \\(x , y\\) \u304c \\(|x| \\leqq 1\\) \u3068 \\(|y| \\leqq 1\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\r\n0 \\leqq x^2 +y^2 -2 x^2 y^2 +2xy \\sqrt{1 -x^2} \\sqrt{1 -y^2} \\leqq 1\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p>\r\n
\\(P = x^2 +y^2 -2 x^2 y^2 +2xy \\sqrt{1 -x^2} \\sqrt{1 -y^2}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nP & = x^2 ( 1 -y^2 ) +y^2 ( 1 -x^2 ) +2xy \\sqrt{1 -x^2} \\sqrt{1 -y^2} \\\\\r\n& = \\left( x \\sqrt{1 -y^2} +y \\sqrt{1 -x^2} \\right)^2 \\geqq 0 \\ .\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n1-P & = x^2y^2 +( 1 -x^2 ) ( 1 -y^2 ) -2xy \\sqrt{1 -x^2} \\sqrt{1 -y^2} \\\\\r\n&= \\left( xy -\\sqrt{1 -x^2} \\sqrt{1 -y^2} \\right)^2 \\geqq 0 \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n0 \\leqq P \\leqq 1 \\ .\r\n\\]\r\n
\u6761\u4ef6\u3088\u308a, \\(x = \\cos \\alpha\\) , \\(y = \\cos \\beta \\ ( 0 \\leqq \\alpha , \\beta \\leqq \\pi )\\) \u3068\u304a\u304f\u3053\u3068\u304c\u3067\u304d\u308b.
\r\n\\(P = x^2 +y^2 -2 x^2 y^2 +2xy \\sqrt{1 -x^2} \\sqrt{1 -y^2}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nP & = \\cos^2 \\alpha ( 1 -\\cos^2 \\beta ) \\\\\r\n& \\qquad +\\cos^2 \\beta ( 1 -\\cos^2 \\alpha ) -2 \\cos \\alpha \\cos \\beta \\sin \\alpha \\sin \\beta \\\\\r\n& = \\cos^2 \\alpha \\sin^2 \\beta +\\cos^2 \\beta \\sin^2 \\alpha -2 \\cos \\alpha \\cos \\beta \\sin \\alpha \\sin \\beta \\\\\r\n& = \\left( \\cos \\alpha \\sin \\beta -\\cos \\beta \\sin \\alpha \\right)^2 \\\\\r\n& = \\sin^2 ( \\alpha -\\beta ) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n0 \\leqq P \\leqq 1 \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(x , y\\) \u304c \\(|x| \\leqq 1\\) \u3068 \\(|y| \\leqq 1\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \u4e0d\u7b49\u5f0f \\[ 0 \\leqq x^2 +y^2 -2 x^2 y^2 +2xy \\sqrt{1 -x^2} … \u7d9a\u304d\u3092\u8aad\u3080