Prove the identity `cos^2 x . tan x = (2sinx)/(secx+cosx+sin^2 x. sec x)`
2 Answers
nikolao | High School Teacher | (Level 1) Adjunct Educator
`cos^2 x . tanx = (2sinx)/(secx+cosx+sin^2 x . sec x)`
`cos^2 x . tanx`
`=> cos^2 x. (sinx/cosx) `
=cosx. sinx ----(1)
`(2.sinx)/(sec x+cosx+sin^2 x sec x)`
`(2.sinx)/((1/cos x) + (cosx) + (sin^2 x)/(cosx))`
`(2.sinx.cosx)/(1+cos^2 x+ sin^2 x)`
`(2. sinx . cosx)/(2)`
sinx.cosx--------(2)
As (1) = (2) So `cos^2 x . tanx = (2sinx)/(secx+cosx+sin^2 x . sec x)`
justaguide | College Teacher | (Level 2) Distinguished Educator
The identity `cos^2 x* tan x = (2sinx)/(secx+cosx+sin^2 x* sec x) ` has to be proved.
`(2sinx)/(secx+cosx+sin^2 x* sec x) `
= `(2sinx)/(1/(cos x)+cosx+sin^2 x*1/(cos x)) `
= `(2sinx*cos x)/(1+cos^2x+sin^2 x) `
= `(2sinx*cos x)/(1+1)`
= `(2sinx*cos x)/2`
= `sinx*cos x`
= `(sin x/cos x)*cos x*cos x`
= `tan x*cos^2x`
This proves that ` cos^2 x* tan x = (2sinx)/(secx+cosx+sin^2 x* sec x)`