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If sec θ = x + 1/4x , x ≠ 0 , then how can (sec θ – tan θ) be expressed as?

If sec θ = x + 1/4x, x ≠ 0, then how can (sec θ – tan θ) be expressed as?

Grade:11

1 Answers

Arun
25750 Points
4 years ago
Given:
 
secϴ = x + 1/4x…….(1)
 
tan²ϴ = sec²ϴ -1
tan²ϴ =  (x + 1/4x)² -1  
 
[From equation 1]
 
tan²ϴ = x² + 1/16x² +½ -1
[ (a+b)² = a² + 2ab +b²]
 
tan²ϴ = x² + 1/16x² - ½
tan²ϴ = (x - 1/4x)²
 
[a² +b²-2ab = (a-b)²]
tanϴ = ±(x - 1/4x)
 
[Taking square roots both sides]
tanϴ = (x - 1/4x) or - (x - 1/4x)
When tanϴ =  (x - 1/4x), then  
secϴ +tanϴ = x +1/4x + x -1/4x = 2x
 
[From equation 1]
 
secϴ +tanϴ = 2x
 
When tanϴ = - (x - 1/4x), then  
secϴ +tanϴ = (x +1/4x) -( x -1/4x )  
 [From equation 1]
 
=  x +1/4x - x + 1/4x   
= 1/4x + 1/4x = 2/4x = 1/2x
Hence secϴ - tanϴ = 2x
 

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