your really close pretty much just have to log both sides
e^-x = e^2 - 2
-x = ln (e^-2 - 2)
-x = 1.68
x = -1.68
3i)
L.H.S. = cosec 2x + cot 2x
L.H.S. = 1 + cos 2x
sin 2x sin 2x
L.H.S. = 1 + cos 2x
sin 2x
L.H.S. = 1 + 2cos^2 x - 1
2sin x cos x
L.H.S. = cos x
sin x
L.H.S. = cot x
L.H.S. = R.H.S.
ii) cosec 2x + cot 2x = 2
therefore cot x = 2
tan x = 1/2
x = 26.6, 26.6+180
x = 26.6, 206.6
5i) (1 + 2x)(1+ax)^2/3 = (1 + 2x)(1 + (2/3)ax + ...)
= 1 + (2/3)ax + ...
+ 2x + (4/3)ax^2 + ...
no coefficient of x
therefore (2/3)a + 2 = 0
(2/3)a = -2
a = -3
ii)(1 + 2x)(1 - 3x)^2/3
= (1 + 2x)(1 - 2/3 * 3x + 2/3 * -1/3 * 1/2! * (-3x)^2 + 2/3 * -1/3 * -4/3 * 1/3! * (-3x)^3 + ...)
= (1 + 2x)(1 - 2x - x^2 - 4/3x^3 + ...)
= 1 - 2x - x^2 - 4/3x^3
+ 2x - 4x^2 - 2x^3 + ...
therefore coefficient of x^3 = -4/3 - 2
= -10/3