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If sec θ = x + 1/4x, prove that sec θ + tan θ = 2x
Answers
Answer: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
secϴ +tanϴ = 1/2x
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Step-by-step explanation:
Step-by-step explanation:
To prove : sec θ + tan θ = 2x
secϴ = x + 1/4x …….(1)
tan²ϴ = sec²ϴ -1
tan²ϴ = (x + 1/4x)² -1
[From (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 ( 1 )]
secϴ +tanϴ = 2x
When tanϴ = - (x - 1/4x), then
secϴ +tanϴ = (x +1/4x) -( x -1/4x )
[From (1)]
= x +1/4x - x + 1/4x
= 1/4x + 1/4x = 2/4x = 1/2x
secϴ +tanϴ = 1/2x