Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.) cos(x + 3π/4) − cos(x − 3π/4) = 1

Answer 1

Greetings! Lets solve this shall we ?

So, we must find all solutions between [0, 2π) of the equation

cos(x + 3π/4) - cos(x - 3π/4) = 1.

Let cos(x + 3π/4) = cosxcos(3π/4) - sinxsin(3π/4) from the Sum & Difference Formula on the trig identity sheet. The same goes for cos(x - 3π/4) = cosxcos(3π/4) + sinxsin(3π/4). Now we can find the solutions by simplification of the equation such that

[cosxcos(3π/4) - sinxsin(3π/4)] - [cosxcos(3π/4) + sinxsin(3π/4)] = 1

cosxcos(3π/4) - cosxcos(3π/4) - sinxsin(3π/4) - sinxsin(3π/4) = 1

-2sinxsin(3π/4) = 1..................sin(3π/4) = √2/2 such that

-2sinx(√2/2) = 1

-2(√2/2)sinx = 1

-√2sinx = 1..................Then divide -√2 such that

sinx = -1/√2...........Rationalize the denominator such that

sinx = -√2/2.............Inverse sine both sides such that

x = sin^-1(-√2/2) = 5π/4 and 7π/4

I hope this helped!