Biết tanx = 2b/(a - c). Giá trị của biểu thức A = cos^2 x + 2bsinxcosx + csin^2x

* Hướng dẫn giải

Đáp án B

$$\begin{gathered} A=a{\cos }^{2}x+2b\sin x.cosx+c{\sin }^{2}x \\ \iff \frac{A}{co{s}^{2}x}=a+2b\tan x+c{\tan }^{2}x \\ \iff A\left(1+{\tan }^{2}x\right)=a+2b\tan x+c{\tan }^{2}x \\ \iff A\left(1+{\left(\frac{2b}{a-c}\right)}^2\right)=a+2b\frac{2b}{a-c}+c{\left(\frac{2b}{a-c}\right)}^2 \\ \iff A\frac{{\left(a-c\right)}^2+{\left(2b\right)}^2}{{\left(a-c\right)}^2}=\frac{a{\left(a-c\right)}^2+4b^2\left(a-c\right)+c4b^2}{{\left(a-c\right)}^2} \\ \iff A\frac{{\left(a-c\right)}^2+{\left(2b\right)}^2}{{\left(a-c\right)}^2}=\frac{a{\left(a-c\right)}^2+4b^{2}a}{{\left(a-c\right)}^2}=\frac{a\left({\left(a-c\right)}^2+4b^2\right)}{{\left(a-c\right)}^2} \\ \iff A=a \end{gathered}$$