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

2 Answers
Aug 31, 2015

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)

Aug 31, 2015

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)