Answer:
[tex]sin^4(a)+cos^4(a)=\frac{41}{50}[/tex]
Step-by-step explanation:
we have
[tex]sin^4(a)+cos^4(a)[/tex]
Complete the square
[tex]sin^4(a)+cos^4(a)=(sin^2(a)+cos^2(a))^2-2sin^2(a)cos^2(a)[/tex]
Remember that
[tex]sin^2(a)+cos^2(a)=1[/tex]
so
[tex]sin^4(a)+cos^4(a)=1-2sin^2(a)cos^2(a)[/tex]
Rewrite
[tex]sin^4(a)+cos^4(a)=1-2(sin(a)cos(a))^2[/tex]
we know that
[tex]sin(2a)=2sin(a)cos(a)\\\\\frac{1}{2}sin(2a)=sin(a)cos(a)[/tex]
In this problem we have
[tex]sin(2a)=\frac{3}{5}[/tex]
so
[tex]\frac{1}{2}sin(2a)=\frac{1}{2}(\frac{3}{5})=\frac{3}{10}[/tex]
[tex]sin(a)cos(a)=\frac{3}{10}[/tex]
substitute
[tex]sin^4(a)+cos^4(a)=1-2(sin(a)cos(a))^2[/tex]
[tex]sin^4(a)+cos^4(a)=1-2(\frac{3}{10})^2[/tex]
[tex]sin^4(a)+cos^4(a)=1-\frac{18}{100}[/tex]
[tex]sin^4(a)+cos^4(a)=\frac{82}{100}[/tex]
simplify
[tex]sin^4(a)+cos^4(a)=\frac{41}{50}[/tex]
