I have a trigonometry problem?

L.H.S.

= tan²x / (secx - 1)²

= (sec²x - 1) / (secx - 1)² …… tan²x = sec²x - 1

= (secx + 1)(secx - 1) / (secx - 1)² …… a² - b² = (a + b)(a - b), and thus sec²x - 1 = (secx + 1)(secx - 1)

= (secx + 1) / (secx - 1) …… eliminate (secx - 1) in both denominator and numerator provided that secx - 1 ≠ 0

= [(1/cosx) + (cosx/cosx)] / [(1/cosx) - (cosx/cosx)] …… secx = 1/cosx and 1 = cosx/cosx

= [(1 + cosx) / cosx] / [(1 - cosx) / cosx]

= (1 + cosx) / (1 - cosx)…… eliminate 1/cosx on both denominator and numerator provided that 1/cosx ≠ 0 and cosx ≠ 0

= R.H.S.

Hence,tan²x / (secx - 1)² = (1 + cosx) / (1 - cosx)

[sec^2(x)-1)]/(secx-1)^2 =
(secx-1)(secx+1)/(secx-1)^2 =
(secx+1)/(secx-1) =
(1/cosx+1)/(1/cosx-1) =
(1+cosx)/(1-cosx)

tan²x / (sec x - 1)²

Replace tan and sec with versions using sin and cos, namely tan x = sin x / cos x and sec x = 1 / cos x:
= (sin²x / cos²x) / (1/cos x - 1)²

Get a common denominator for the second pair of terms:
= (sin²x / cos²x) / (1/cos x - cos x/cos x)²
= (sin²x / cos²x) / ((1 - cos x)/cos x)²

Distribute the exponent in the denominator:
= (sin²x / cos²x) / ((1 - cos x)²/cos²x)

Change it to a multiplication by the reciprocal:
= (sin²x / cos²x) * (cos²x)/(1 - cos x)²

Cancel out cos²x:
= sin²x / (1 - cos x)²

Remember that sin²x + cos²x = 1 --> sin²x = 1 - cos²x, so replace the numerator:
= (1 - cos²x) / (1 - cos x)²

The numerator is a difference of squares --> a² - b² = (a - b)(a + b) where a = 1 and b = cos x:
= (1 - cos x)(1 + cos x) / (1 - cos x)²
= (1 - cos x)(1 + cos x) / ((1 - cos x)(1 - cos x))

Cancel 1 - cos x from top and bottom:
(1 + cos x) / (1 - cos x)

LHS = (sec^2x - 1) /(sec x - 1)^2 = (sec x + 1 ) (sec x - 1) /(sec x - 1)^2 = (sec x + 1 )/(sec x - 1 ) = (1+cos x) / (1- cos x) = RHS

tan^2x/(secx - 1)^2 =
tan^2x/(sec^2x - 2secx + 1) =
sin^2x/(cos^2x(sec^2x - 2secx + 1)) =
sin^2x/(1 - 2cosx + cos^2x) =
sin^2x/(1 - cosx)^2 =
(1 - cos^2x)/(1 - cosx)^2 =
(1 + cosx)(1 - cosx)/(1 - cosx)^2 =
(1 + cosx)/(1 - cosx)

tan^2 x /(sec x -1)^2
= tan^2 x / (sec^2 x - 2 sec x +1)
= (sec^2 x -1) /(sec^2 x -2 sec x +1)
divide the numerator and the denominator by sec^2 x
(1 - cos^2 x) /(1 - 2 cos x + cos^2 x)
= (1 -cos^2 x)/(1-cos x)^2
= (1-cos x)(1+cos x) /((1-cos x)(1-cos x))
= (1+cos x)/(1-cos x)