a.
\(\left(sina+cosa\right)^2=sin^2a+cos^2a+2sina.cosa=1+sin2a\)
b.
\(cos^4a-sin^4a=\left(cos^2a-sin^2a\right)\left(cos^2a+sin^2a\right)=cos2a.1=cos2a\)
c.
\(sin^4a+cos^4a+\dfrac{1}{2}sin^22a=sin^4a+cos^4a+\dfrac{1}{2}\left(2sina.cosa\right)^2\)
\(=sin^4a+cos^4a+\dfrac{1}{2}.4sin^2a.cos^2a=sin^4a+cos^4a+2sin^2a.cos^2a\)
\(=\left(sin^2a+cos^2a\right)^2=1^2=1\)
d.
\(sina-\sqrt{3}cosa=2\left(\dfrac{1}{2}sina-\dfrac{\sqrt{3}}{2}cosa\right)\)
\(=2\left(sina.cos\dfrac{\pi}{3}-cosa.sin\dfrac{\pi}{3}\right)=2sin\left(a-\dfrac{\pi}{3}\right)\)
e.
\(sin^6a+cos^6a=\left(sin^2a\right)^3+\left(cos^2a\right)^3\)
\(=\left(sin^2a+cos^2a\right)^3-3sin^2a.cos^2a\left(sin^2a+cos^2a\right)\)
\(=1^3-3sin^2a.cos^2a.1\)
\(=1-3sin^2a.cos^2a=1-\dfrac{3}{4}.\left(2sina.cosa\right)\)
\(=1-\dfrac{3}{4}sin^22a=1-\dfrac{3}{4}\left(\dfrac{1-cos4a}{2}\right)\)
\(=1-\dfrac{3}{8}+\dfrac{3}{8}cos4a=\dfrac{5}{8}+\dfrac{3}{8}cos4a\)
