1.
Xét $2\cos a(1-\cos a)=(\sin a+\cos a-1)(\sin a-(\cos a-1))$
$\to 2\cos a-2\cos^2a=\sin^2a-(\cos a-1)^2$
$\to 2\cos a-2\cos^2a=\sin^2a-\cos^2a+2\cos a-1$
$\to \sin^2a-\cos^2a+2\cos^2a-1=0$
$\to \sin^2a+\cos^2a=1$ (luôn đúng)
Vậy $\dfrac{\sin a+\cos a-1}{1-\cos a}=\dfrac{2\cos a}{\sin a-\cos a+1}$
2.
$\cos^2a(1+\cos^2a)$ không thể rút gọn thêm.
3.
$\sin a+\cos a=\dfrac{7}{5}$
$\to \sin^2a+\cos^2a+2\sin a\cos a=\dfrac{49}{25}$
$\to \sin a\cos a=\dfrac{ \dfrac{49}{25}-1}{2}=\dfrac{12}{25}$
$\to \tan a+\cot a=\dfrac{\sin a}{\cos a}+\dfrac{\cos a}{\sin a}=\dfrac{\sin^2a+\cos^2a}{\sin a\cos a}=\dfrac{1}{\sin a\cos a}=\dfrac{25}{12}$
Mà $\tan a.\cot a=1$
Hai số $\tan a$, $\cot a$ là nghiệm pt:
$t^2-\dfrac{25}{12}t+1=0$
$\to t=\dfrac{4}{3}$ hoặc $t=\dfrac{3}{4}$
Vậy $\Big(\tan a;\cot a\Big)=\Big(\dfrac{3}{4};\dfrac{4}{3}\Big), \Big(\dfrac{4}{3};\dfrac{3}{4}\Big)$
