if sec theta = x + 1/4x, prove thatvsec theta + tan theta = 2x or 1/2x.
Answers
Step-by-step explanation:]
Sec A = x + 1/4x
As, 1 + Tan²A = Sec²A
Tan²A = (x+ 1/4x)² - 1
= x² +2x*1/4x + 1 /16x² -1
= x² + 1/16² - 1/2
= (x-1/4x)²
Tan²A = x-1/4x or Tan²A = -(x-1/4x)
Substitute the value of SecA and TanA in the given equation Sec A +Tan A
L.H.S = SecA + TanA
= x + 1/4x - x + 1/4x
= 2/4x
=R.H.S
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Given that secθ = x+1/4x
1+tan²θ = sec²θ
or tan²θ = sec²θ – 1
on expanding
tan²θ = ( x+1/4x)² -1
or tan²θ = ( x²+1/16x² + 1/2 -1 )
or tan²θ = (x² +1/16x² -1/2 )
or tan²θ = x² +1/16x²-1/2
or tan²θ = (x-1/4x)^2
or tan²θ = +(x-1/4x) or – (x-1/4x)
when tanθ = (x-1/4x) we get
secθ+tanθ = x+1/4x + x-1/4x = 2x
when tanθ = -(x-1/4x)
secθ+tanθ = (x+1/4x) – (x-1/4x) = 1/2x
Hence
secθ+tanθ = (x+1/4x) – (x-1/4x) = 1/2x
and secθ+tanθ = x+1/4x + x-1/4x = 2x