plz help me with all the stepsprove that sin^6(x)+cos^6 (x) = 1- 3/4 sin^2(2x)i really need help...β
Accepted Solution
A:
Step-by-step explanation:Since the identity is true whether the angle x is measured in degrees, radians, gradians (indeed, anything else you care to concoct), Iβll omit the βdegreesβ sign.
Using the binomial theorem, (a+b)3=a3+3a2b+3ab2+b3
βa3+b3=(a+b)3β3a2bβ3ab2=(a+b)3β3(a+b)ab
Substituting a=sin2(x) and b=cos2(x), we have:
sin6(x)+cos6(x)=(sin2(x)+cos2(x))3β3(sin2(x)+cos2(x))sin2(x)cos2(x)
Using the trigonometric identity cos2(x)+sin2(x)=1, your expression simplifies to:
sin6(x)+cos6(x)=1β3sin2(x)cos2(x)
From the double angle formula for the sine function, sin(2x)=2sin(x)cos(x)βsin(x)cos(x)=0.5sin(2x)
Meaning the expression can be rewritten as:
sin6(x)+cos6(x)=1β0.75sin2(2x)=1β34sin2(2x)