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tanx+2tan2x+4tan4x+8cot8x . what is the value of this

tanx+2tan2x+4tan4x+8cot8x . what is the value of this
 

Grade:11

1 Answers

Arun
25758 Points
4 years ago
First, its important to find the value of tan x – cot x. So, tan x – cot x = tan x - 1/tan x 
= tan² x-1/tan x
 = -2 (1-tan² x/ 2tan x) 
= -2 ( 1/2tan x ÷ 1- tan² x)
 = -2× 1/tan² x = 2cot 2x ....(1) .
 Now— 
tan x + 2 tan 2x + 4 tan 4x + 8 cot 8x
 = tan x – cot x + 2 tan 2x + 4 tan 4x + 8 cot 8x + cot x
= –2 cot 2x + 2 tan 2x + 4 tan 4x + 8 cot 8x + cot x
 = 2 (tan 2x – cot 2x) + 4 tan 4x + 8 cot 8x + cot x
= 2 (– 2 cot 4x) + 4 tan 4x + 8 cot 8x + cot x
= 4 (tan 4x – cot 4x) + 8 cot 8x + cot x
= 4 (– 2 cot 8x) + 8 cot 8x + cot x
= – 8 cot 8x + 8 cot 8x + cot x
= cot x . 
Hence, tan x + 2 tan 2x + 4 tan 4x + 8 cot 8x = cot x

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