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)#