Đáp án:
$\left[ \begin{array}{l}\cot\alpha=\dfrac{4}{3}\\\cot\alpha=\dfrac{3}{4}\end{array} \right.$
Giải thích các bước giải:
$\sin\alpha-\cos\alpha=\dfrac{1}{5}$
$⇒\left(\sin\alpha-\cos\alpha\right)^2=\dfrac{1}{25}$
$⇒\sin^2\alpha-2\sin\alpha.\cos\alpha+\cos^2\alpha=\dfrac{1}{25}$
$⇒1-2\sin\alpha.\cos\alpha=\dfrac{1}{25}$
$⇒2\sin\alpha.\cos\alpha=\dfrac{24}{25}$
$⇒\sin\alpha.\cos\alpha=\dfrac{12}{25}$
$⇒\dfrac{\sin\alpha.\cos\alpha}{\sin^2\alpha}=\dfrac{12}{25\sin^2\alpha}$
$⇒\dfrac{\cos\alpha}{\sin\alpha}=\dfrac{12}{25}.\dfrac{\sin^2\alpha+\cos^2\alpha}{\sin^2\alpha}$
$⇒\cot\alpha=\dfrac{12}{25}.\left(\dfrac{\sin^2\alpha}{\sin^2\alpha}+\dfrac{\cos^2\alpha}{\sin^2\alpha}\right)$
$⇒25\cot\alpha=12+12\cot^2\alpha$
$⇒12\cot^2\alpha-25\cot\alpha+12=0$
$⇒\left[ \begin{array}{l}\cot\alpha=\dfrac{4}{3}\\\cot\alpha=\dfrac{3}{4}\end{array} \right.$
Vậy $\left[ \begin{array}{l}\cot\alpha=\dfrac{4}{3}\\\cot\alpha=\dfrac{3}{4}\end{array} \right.$.
