n2tcost - cos2tsint = 0 on the interval 0≤t≥2Π

Question

Solve algebraically, exact solutions only, sin2tcost - cos2tsint = 0 on the interval 0≤t≥2Π

Answer 1

sin(2t)cost - cos(2t)sint = 0

 

2sintcostcost - (1-2sin²t)sint = 0

 

2sintcos²t - sint + 2sin³t = 0

 

sint(2cos²t + 2sin²t -1) = 0

 

sint[2(cos²t + sin²t) - 1] = 0   Recall that cos²t + sin²t = 1

 

sint[2(1) - 1] = 0       sint = 0

 

                                     t = 0, π, 2π

 

Answer 2

assuming sin²tcost-cos²tsint=0

factor

sintcost(sint-cost)=0

sintcost=0 or sint-cost=0

sintcost=0 means sint=0 or cost=0

sint=0 when t=0º or t=180º (look on the unit circle)

cost=0 when t=90º or t=270º (look on the unit circle)

sint-cost=0

sint=cost

t=45º or t=225º (look on the unit circle)

t=0, 180, 90, 270, 45, and 225 degrees