Question

[nanako]

Đơn giản các biểu thức sau:

a) sin a.\(\sqrt{1+tan^2a}\)

b) \(\frac{1-cos^2x}{1-sịn^2x}+tanx.cotx\)

c) \(\frac{1-4sin^2x.cos^2x}{\left(sinx+cosx\right)^2}\)

d) sin(90^o-x)+cos(180⁰-x)+sin²x(1+tan²x)-tan²x

Answer 1

[Nguyễn Việt Lâm]

\(sina\sqrt{1+\frac{sin^2a}{cos^2a}}=sina\sqrt{\frac{cos^2a+sin^2a}{cos^2a}}=\frac{sina}{\left|cosa\right|}=\pm tana\)

\(\frac{1-cos^2x}{1-sin^2x}+tanx.cotx=\frac{sin^2x}{cos^2x}+\frac{sinx}{cosx}.\frac{cosx}{sinx}=tan^2x+1=\frac{1}{cos^2x}\)

\(\frac{1-4sin^2xcos^2x}{\left(sinx+cosx\right)^2}=\frac{\left(1-2sinx.cosx\right)\left(1+2sinx.cosx\right)}{sin^2x+cos^2x+2sinx.cosx}=\frac{\left(1-sin2x\right)\left(1+2sinx.cosx\right)}{1+2sinx.cosx}=1-2sinx\)

\(sin\left(90-x\right)+cos\left(180-x\right)+sin^2x\left(1+tan^2x\right)-tan^2x\)

\(=cosx-cosx+sin^2x.\frac{1}{cos^2x}-tan^2x=tan^2x-tan^2x=0\)