Đáp án + Giải thích các bước giải:

Bài 3:

a) $VT = \dfrac{\sin^3 \alpha + \cos^3 \alpha}{\sin \alpha + \cos \alpha}$

$= \dfrac{(\sin \alpha + \cos \alpha)(\sin^2 \alpha - \sin \alpha \cos \alpha + \cos^2 \alpha)}{\sin \alpha + \cos \alpha}$

$= \dfrac{(\sin \alpha + \cos \alpha)(1 - \sin \alpha \cos \alpha)}{\sin \alpha + \cos \alpha}$

$= 1 - \sin \alpha \cos \alpha = VP$

$\Rightarrow \text{ĐPCM}$

b) $VT = \dfrac{1 + \tan \alpha}{1 - \tan \alpha} + \dfrac{1 + \cot \alpha}{1 - \cot \alpha}$

$= \dfrac{(1 + \tan \alpha)(1 - \cot \alpha) + (1 + \cot \alpha)(1 - \tan \alpha)}{(1 - \tan \alpha)(1 - \cot \alpha)}$

$= \dfrac{1 + \tan \alpha - \cot \alpha - \tan \alpha \cot \alpha + 1 + \cot \alpha - \tan \alpha - \cot \alpha \tan \alpha}{(1 - \tan \alpha)(1 - \cot \alpha)}$

$= \dfrac{1 + \tan \alpha - \cot \alpha - 1 + 1 + \cot \alpha - \tan \alpha - 1}{(1 - \tan \alpha)(1 - \cot \alpha)}$

$= \dfrac{0}{(1 - \tan \alpha)(1 - \cot \alpha)}$

$= 0 = VP$

$\Rightarrow \text{ĐPCM}$