Chứng minh các đồng nhất thức: `(sinx+sin(x/2))/(1+cosx+cos(x/2))=tan(x/2)` 04/09/2021 Bởi Valentina Chứng minh các đồng nhất thức: `(sinx+sin(x/2))/(1+cosx+cos(x/2))=tan(x/2)`
$\quad \dfrac{\sin x + \sin\left(\dfrac x2\right)}{1 + \cos x + \cos\left(\dfrac x2\right)}$ $= \dfrac{2\sin\left(\dfrac x2\right).\cos\left(\dfrac x2\right) + \sin\left(\dfrac x2\right)}{1 + 2\cos^2\left(\dfrac x2\right) – 1 + \cos\left(\dfrac x2\right)}$ $= \dfrac{\sin\left(\dfrac x2\right)\left[2\cos\left(\dfrac x2\right)+1\right]}{\cos\left(\dfrac x2\right)\left[2\cos\left(\dfrac x2\right)+1\right]}$ $=\dfrac{\sin\left(\dfrac x2\right)}{\cos\left(\dfrac x2\right)}$ $=\tan\left(\dfrac x2\right)$ Bình luận
Đáp án:
Giải thích các bước giải:
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$\quad \dfrac{\sin x + \sin\left(\dfrac x2\right)}{1 + \cos x + \cos\left(\dfrac x2\right)}$
$= \dfrac{2\sin\left(\dfrac x2\right).\cos\left(\dfrac x2\right) + \sin\left(\dfrac x2\right)}{1 + 2\cos^2\left(\dfrac x2\right) – 1 + \cos\left(\dfrac x2\right)}$
$= \dfrac{\sin\left(\dfrac x2\right)\left[2\cos\left(\dfrac x2\right)+1\right]}{\cos\left(\dfrac x2\right)\left[2\cos\left(\dfrac x2\right)+1\right]}$
$=\dfrac{\sin\left(\dfrac x2\right)}{\cos\left(\dfrac x2\right)}$
$=\tan\left(\dfrac x2\right)$