Question

If tan theta= x-1/4x,then sec theta - tan theta=

Answer 1

Given: tanθ = x - 1/4x

To find: The value of secθ - tanθ.

Solution:

• We have given tanθ = x - 1/4x, so:

                  tanθ = ( 4x² - 1 ) / 4x

• We know that sec²θ = 1 + tan²θ                            

• So:

                  sec²θ = 1 +  (( 4x²- 1 ) / 4x)²

                  sec²θ = ( 16x² + 16x^4 - 8x² ) / 16x²

• We can write it in simplified form as:

                  sec²θ = (4x² + 1)² / (4x)²

                  secθ = ±( 4x² + 1 ) / 4x

• So now putting values in secθ - tanθ, we get:

• If secθ is positive, then :

                  secθ - tanθ = ( 4x² + 1 ) / 4x - ( 4x² - 1 ) / 4x

                  secθ - tanθ = 4x² + 1 - 4x² + 1 / 4x

                  secθ - tanθ = 2/4x

                  secθ - tanθ = 1/2x

• If secθ is negative, then :

                  secθ - tanθ = -( 4x² + 1 ) / 4x - ( 4x² - 1 ) / 4x

                  secθ - tanθ = -4x² - 1 - 4x² + 1 / 4x

                  secθ - tanθ = -8x²/4x

                  secθ - tanθ = -2x

Answer:

             So the value of secθ - tanθ is 1/2x and -2x.