Основные тригонометрические формулы. Задача 17math100admin44242025-03-21T19:59:42+03:00
Задача 17. Упростите выражение \(\dfrac{{1-{{\cos }^4}x-{{\sin }^4}x}}{{{{\left( {{{\sin }^2}x-{{\cos }^2}x} \right)}^2}\,-{{\sin }^4}x-{{\cos }^4}x}}\)
Решение
\(\dfrac{{1-{{\cos }^4}x-{{\sin }^4}x}}{{{{\left( {{{\sin }^2}x-{{\cos }^2}x} \right)}^2}-{{\sin }^4}x-{{\cos }^4}x}} = \dfrac{{\left( {1-{{\cos }^2}x} \right)\left( {1 + {{\cos }^2}x} \right)-{{\sin }^4}x}}{{{{\sin }^4}x-2{{\sin }^2}x{{\cos }^2}x + {{\cos }^4}x-{{\sin }^4}x-{{\cos }^4}x}} = \)
\( = \dfrac{{{{\sin }^2}x\left( {1 + {{\cos }^2}x} \right)-{{\sin }^4}x}}{{-2{{\sin }^2}x{{\cos }^2}x}} = \dfrac{{{{\sin }^2}x\left( {1 + {{\cos }^2}x-{{\sin }^2}x} \right)}}{{-2{{\sin }^2}x{{\cos }^2}x}} = \)
\( = \dfrac{{1-{{\sin }^2}x + {{\cos }^2}x}}{{-2{{\cos }^2}x}} = \dfrac{{{{\cos }^2}x + {{\cos }^2}x}}{{-2{{\cos }^2}x}} = \dfrac{{2{{\cos }^2}x}}{{-2{{\cos }^2}x}} = -1.\)
Ответ: \(-1.\)