Question

Verify the following identities: (a) `2sin^2(2x) + cos4x=1` (b) `tanx+co x =2sec2x` (c) `tan3x =[tanx(3-tan^2x)]/ (1-3tan^2x)`?

Answer 1

Verify:
1. 2sin^2 (2x) + cos 4x = 1
2. tan x + cot x = 2csc 2x

Explanation:

1. `2sin^2 (2x) + cos 4x = 2sin^2 (2x) + (1 - 2sin^2 (2x) = 1`
   Reminder of trig identity: `cos 2a = 1 - 2sin^2 a`

2. `tan x + cot x = sin x/cos x + cos x/sin x =`
   = `(sin^2 x + cos^2 x)/(sin x.cos x) = 1/(sin x.cos x) =`
   `= 2/(sin 2x) = 2csc (2x)`

Answer 2

Verify: `tan 3x = ((tan x)(3 - tan^2 x))/(1 - 3tan^2 x)`

Explanation:

Reminder: Trig identity `tan 2a = (2tan a)/(1 - tan^2 a) (1)`, and
Trig Identity: `tan (a + b) = (tan a + tan b)/(1 - tan a.tan b) (2)`

`tan 3x = tan (2x + x) = (tan x + tan 2x)/(1 - tan x.tan 2x) (3)`
Substitute into (3) the value of tan 2x from Identity (1)

Develop the numerator:#(tan x + (2tan x)/(1 - tan^2 x)) = (3tan x - tan^3 x)/(1 - tan^2 x) =#
`= ((tan x)(3 - tan^2 x))/(1 - tan^2 x)` (4)
Develop the denominator: #(1 - tan x.tan 2x) = =1 - (tan x(2tan x))/(1 - tan^2 x)# =
`(1 - tan^2 x - 2tan^2 x)/(1 - 2tan^2 x) = (1 - 3tan^2 x)/(1 - tan^2 x)` (5)

`tan 3x = ((4))/((5)) = (tan x(3 - tan^2 x))/(1 - 3tan^2 x)`