Question

решите уравнение
$2^{tg(x-frac{Pi}{4})} - 2 * 0.25^{frac{sin^{2}(x-frac{Pi}{4})}{cos2x}} - 1 = 0$

Answer 1

Решение приложено

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Answer 2

y = -1

Answer 3

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^bigg{frac{sin^{2}left(x - frac{pi}{4} right)}{cos 2x} } - 1 = 0$

ОДЗ:

$left{begin{matrix}x - dfrac{pi}{4} neq dfrac{pi}{2} + pi n, n in Z \ \ 2x neq dfrac{pi}{2} + pi k, k in Z end{matrix}right.$

$left{begin{matrix}x neq dfrac{3pi}{4} + pi n, n in Z \\x neq dfrac{pi}{4} + dfrac{pi k}{2}, k in Z end{matrix}right.$

$x neq dfrac{pi}{4} + dfrac{pi k}{2}, k in Z$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ left(sin x cos frac{pi}{4} - cos x sin frac{pi}{4} right)^{2}}{ cos 2x }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ left(frac{sqrt{2}}{2} sin x - frac{sqrt{2}}{2} cos x right)^{2}}{ cos 2x}}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ 0,5sin^{2}x - sin x cos x + 0,5 cos^{2}x}{ cos 2x }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ 1 - 2sin x cos x}{ 2cos 2x }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ 1 - 2text{tg} x cos^{2} x}{ 2(2cos^{2} x - 1) }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ 1 - bigg{frac{2 text{tg} x}{1 + text{tg}^{2}x}} }{ bigg{frac{4}{1 +text{tg}^{2}x}-2} }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{ 1 - 2text{tg}x + text{tg}^{2}x}{ 2(1 - text{tg}^{2}x) }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{1}{2} cdot frac{(1 - text{tg}x)^{2}}{(1 - text{tg}x)(1 + text{tg}x)} }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{frac{1}{2} cdot frac{1 - text{tg}x}{1 + text{tg}x} }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{-frac{1}{2} cdot frac{text{tg}x-1}{1 + text{tg}x} }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{-frac{1}{2} cdot frac{text{tg}x-text{tg}bigg{frac{pi}{4}} }{1 + text{tg}x cdot text{tg}bigg{frac{pi}{4}}} }}} - 1 = 0$

$2^bigg{text{tg}left(x - frac{pi}{4} right )} - 2 cdot 0,25^{bigg{-frac{1}{2} cdot text{tg}left(x - frac{pi}{4} right )}} - 1 = 0$

Замена: $text{tg}left(x - dfrac{pi}{4} right ) = t$

$2^{t} - 2 cdot 0,25^bigg{-frac{1}{2} cdot t } - 1 = 0$

$2^{t} - 2cdot 2^{t} -1 = 0$

$-2^{t} - 1 = 0$

$2^{t} = -1$

$t in varnothing$

Ответ: нет действительных корней.