Find all solutions to the equation cos x cos(3x) - sin x sin(3x) = 0

Solution:

Given equation is

cos x cos(3x) - sin x sin(3x) = 0

To find: x

We know that :

cos(A + B) = cos AcosB - sinAsinB

Hence we can say that:

= cos x cos(3x) - sin x sin(3x)

= cos (x + 3x)

= cos4x

Now,

cos 4x = 0

The cosine of a function is zero at π/2 and 3π/2 (between 0 and 2π)

Therefore :

cos4x = cosπ/2

⇒ 4x = π/2

x = π/8 --- (1)

cos 4x = cos(3π/2)

⇒ 4x = 3π/2

x = 3π/8 --- (2)

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Find all solutions to the equation cos x cos(3x) - sin x sin(3x) = 0

Summary:

The solutions to the equation cos x cos(3x) - sin x sin(3x) = 0 are x = π /8 & 3 π/8