Use the trig identity: color(blue)(sin (a + b) + sin (a - b) = 2sin a*cos b)sin(a+b)+sin(a−b)=2sina⋅cosb
f(x) = 2*sin x*cos (pi/4) - 1 = 0f(x)=2⋅sinx⋅cos(π4)−1=0 ,
since color(blue)(cos (pi/4) = (sqrt2)/2cos(π4)=√22
f(x) = (sqrt2*sin x) - 1 = 0f(x)=(√2⋅sinx)−1=0
sin x = 1/sqrt2 = (sqrt2)/2 --> sinx=1√2=√22−→
color(red)(x = pi/4 and 3pi/4x=π4and3π4 (inside interval 0 - 2pi0−2π)
Check:
x = pi/4 --> x + pi/4 = pi/2 --> sin (x + pi/4) = 1; cos (x + pi/4) = 0 --> f(x) = 1 - 1 = 0x=π4−→x+π4=π2−→sin(x+π4)=1;cos(x+π4)=0−→f(x)=1−1=0. Correct.
x = 3pi/4 --> (x + pi/4) = pi --> sin pi = 0; cos pi = -1 --> f(x) = 1 - 1 = 0.x=3π4−→(x+π4)=π−→sinπ=0;cosπ=−1−→f(x)=1−1=0. Correct
