(2 - sec²x) / (sec²x + 2 tan x)

= (2 - 1/cos²x) / (1/cos²x + 2sinx/cosx)

= (2cos²x - 1) / (1 + 2sinx cosx)

(Multiply both numerator and denominator by cos²x)

= (cos²x - sin²x) / (cos²x + sin²x + 2sinx cosx)

(Recall that cos2x = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x and cos²x + sin²x = 1)

= (cosx + sin x)(cosx - sinx) / (cosx + sinx)²


(Recall that a² - b² = (a + b)(a - b) and (a + b)² = a² + b² + 2ab.

In this case your a = cosx and b = sinx for the numerator. For the denominator, a and b are interchangeable)

= (cosx - sinx)/(cosx + sinx)

(Cancelling out common factor cosx + sinx)

(Proved)

Alternative working :


(2 - sec²x) / (sec²x + 2 tan x)

= (2 - (1 + tan²x)) / (1 + tan²x + 2tanx)

= (1 - tan²x) / (1 + tanx)²

= (1 - tanx)(1 + tanx) / (1 + tanx)²

= (1 - tanx)/(1 + tanx)

= (1 - sinx/cosx)/(1 + sinx/cosx)

= (cosx - sinx)/(cosx + sinx)

(Multiply both numerator and denominator by cosx keeps the expression equivalent.

It's basically changing the denominator of fractions.

Eg. 3/5 × 6/6 = 18/30