Question

(sin⁡(a+cos⁡a)^2- 1)/(ctg a-sin⁡a ×cos⁡a )=2tg^2 a

Answer 1

$\frac{(sina+cosa)^2-1}{ctga-sina\cdot cosa}= \frac{(sin^2a+cos^2a)+2\, sina\, cosa-1}{\frac{cosa}{sina}-sina\, cosa}= \frac{1+2sina\, cosa-1}{\frac{cosa-sin^2a\, cosa}{cosa} }=\\\\= \frac{2\, sin^2a\, cosa}{cosa(1-sin^2a)}= \frac{2sin^2a}{cos^2a} =2tg^2a\; ;\\\\2tg^2a=2tg^2a$

Answer 2

Давайте пошагово рассмотрим данное уравнение:

\[\frac{\sin^2(a + \cos a) - 1}{\cot a - \sin a + \cos a} = 2\tan^2 a\]

1. Раскроем квадрат синуса в числителе:

\[\frac{\sin^2(a + \cos a) - 1}{\cot a - \sin a + \cos a}\]

\[= \frac{(\sin(a)\cos(\cos a) + \cos(a)\sin(\cos a))^2 - 1}{\cot a - \sin a + \cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a)\sin^2(\cos a) - 1}{\cot a - \sin a + \cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a)(1 - \cos^2(\cos a)) - 1}{\cot a - \sin a + \cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a) - \cos^4(\cos a) - 1}{\cot a - \sin a + \cos a}\]

2. Раскроем квадрат котангенса в знаменателе:

\[\frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a) - \cos^4(\cos a) - 1}{\cot a - \sin a + \cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a) - \cos^4(\cos a) - 1}{\frac{\cos a}{\sin a} - \sin a + \cos a}\]

3. Приведем общий знаменатель:

\[\frac{\sin^2(a)\cos^2(\cos a) + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a) + \cos^2(a) - \cos^4(\cos a) - 1}{\frac{\cos a}{\sin a} - \sin a + \cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \sin a\sin a + \cos a\sin a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a(1 - \sin^2 a) + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \cos a\sin^2 a + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a(1 - \sin^2 a) + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \cos a\sin^2 a + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \cos a\sin^2 a + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a(1 - \sin^2 a) + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \cos a\sin^2 a + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a(1 - \sin^2 a) + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a + \cos^2(a)\sin a - \cos^4(\cos a)\sin a - \sin a}{\cos a - \cos a\sin^2 a + \sin a\cos a}\]

\[= \frac{\sin^2(a)\cos^2(\cos a)\sin a + 2\sin(a)\cos(a)\sin(\cos a)\cos(\cos a)\sin a +