Its given that sex(x)-tan(x)=-5, Show that sec(x)+tan(x)=-0.2. Hence find the exact value of cos(x).Hence solve the equation sec(2x-70) - tan(2x-70) = -5 in the range -90<=x<=90 MPC3 AQA 2016 Q9

From the first 2 equations we can see that they are the difference of 2 squares from sec²(x) - tan²(x).This factorises to (sex(x)-tan(x)) * (sec(x)+tan(x)) (1), this can be rearranged to find its equal to 1 (using the identity sec²(x) = 1 + tan²(x)). From this we can substitue back into (1) to find -5 * (sec(x) + tan(x)) = 1, hence sec(x) + tan(x) = -1/5 or -0.2.Combine the first two equations to give 2* sec(x) = -5.2, we can rearrange to get 1/cox(x) = -2.6, so cox(x) is -1/2.6.Finally, as cos(x) is a solution to the above equations we can use the substitution to find that cos(2x-70) = -1/2.6, giving us values 2x-70 = -112.6 and -247.4 and so we rearrange to get x = -21.3 and -88.7 which are the only values present in the range specified.