sec theta = X+1/4x , prove that sec theta + tan theta =2x or 1/2x
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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
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