a.
\(sin^4a+cos^4a=\left(sin^2a+cos^2a\right)^2-2sin^2a.cos^2a\)
\(=1-\dfrac{1}{2}\left(2sina.cosa\right)^2=1-\dfrac{1}{2}sin^22a\)
\(=1-\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{2}cos4a\right)=\dfrac{3}{4}+\dfrac{1}{4}cos4a\)
b.
\(sin^6a+cos^6a=\left(sin^2a+cos^2a\right)^2-3sin^2a.cos^2a\left(sin^2a+cos^2a\right)\)
\(=1-3sin^2a.cos^2a=1-\dfrac{3}{4}\left(2sina.cosa\right)^2\)
\(=1-\dfrac{3}{4}sin^22a=1-\dfrac{3}{4}\left(\dfrac{1}{2}-\dfrac{1}{2}cos4a\right)\)
\(=\dfrac{5}{8}+\dfrac{3}{8}cos4a\)
e.
\(\dfrac{cos2a}{1+sin2a}=\dfrac{cos^2a-sin^2a}{sin^2a+cos^2a+2sina.cosa}=\dfrac{\left(cosa-sina\right)\left(cosa+sina\right)}{\left(sina+cosa\right)^2}=\dfrac{cosa-sina}{cosa+sina}\)
f.
\(cotx+tanx=\dfrac{cosx}{sinx}+\dfrac{sinx}{cosx}\)
\(=\dfrac{sin^2x+cos^2x}{sinx.cosx}=\dfrac{1}{sinx.cosx}\)
\(=\dfrac{2}{2sinx.cosx}=\dfrac{2}{sin2x}\)
c.
\(sina.cos^5a-cosa.sin^5a=sina.cosa\left(cos^4a-sin^4a\right)\)
\(=sina.cosa\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)\)
\(=\dfrac{1}{2}.\left(2sina.cosa\right)\left(cos^2a-sin^2a\right).1\)
\(=\dfrac{1}{2}sin2a.cos2a=\dfrac{1}{4}sin4a\)
d.
\(=\left(cos^4a-sin^4a\right)\left(cos^4a+sin^4a\right)\)
\(=\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)\left(1-\dfrac{1}{2}sin^22a\right)\) (sử dụng kết quả câu a)
\(=cos2a\left(1-\dfrac{1}{2}sin^22a\right)=cos2a-\dfrac{1}{2}cos2a.sin2a.sin2a\)
\(=cos2a-\dfrac{1}{4}sin4a.sin2a\)
g.
\(cotx-tanx=\dfrac{cosx}{sinx}-\dfrac{sinx}{cosx}\)
\(=\dfrac{cos^2x-sin^2x}{sinx.cosx}=\dfrac{cos2x}{\dfrac{1}{2}sin2x}\)
\(=2.cot2x\)
h.
\(\dfrac{sin2x}{1+cos2x}=\dfrac{2sinx.cosx}{1+2cos^2x-1}\)
\(=\dfrac{2sinx.cosx}{2cos^2x}=\dfrac{sinx}{cosx}\)
\(=tanx\)
i.
\(\dfrac{1-cos2x}{1+cos2x}=\dfrac{1-\left(1+2sin^2x\right)}{1+2cos^2x-1}\)
\(=\dfrac{2sin^2x}{2cos^2x}=tan^2x\)
j.
\(cos^3a.sina-sin^3a.cosa=sina.cosa\left(cos^2a-sin^2a\right)\)
\(=\dfrac{1}{2}sin2a.cos2a\)
\(=\dfrac{1}{4}sin4a\)
