Factorise: 2 cos2x−cos x−3

To factorise the expression 2cos(2x) - cos(x) - 3, we first notice that this expression can be simplified by using the double angle formula for cosine:

cos(2x) = 2cos^2(x) - 1

Substitute this into our original expression to get:

2(2cos^2(x) - 1) - cos(x) - 3
4cos^2(x) - 2 - cos(x) - 3
4cos^2(x) - cos(x) - 5

Now we have a quadratic expression that we can factorise. To factorise 4cos^2(x) - cos(x) - 5, we look for two values that multiply to -20 (4*-5) and add up to -1 (the coefficient of the middle term, -cos(x)). The two values are -5 and 4.

Therefore, the factorised expression is:

(4cos(x) + 5)(cos(x) - 1)