(cot(theta) - 1)/(cot(theta) + 1) = (1 - sin2(theta))/(cos2(theta))

Dear friend please take theta = x ( it is easy in solving )

Cot x - 1 / cotx + 1

( cos x /sinx - 1 ) /( cosx/sinx + 1 )

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

Cancelling sinx from numerator and denominatior

(cosx - sinx) /( cosx + sinx)

multiply numberatior and denominator by cosx - sinx

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

(cosx - sinx )^2/cos^2x - sin^x

cos^2x + sin^2x - 2sinx cosx / cos2x

cos^2x - sin^2x = cos2x ( identitiy used )

1 - sin2x / cos2x

cos^2x + sin^2 x = 1

2 sinx cosx = sin2x

Now you may replace x by theta

1 - sin2theta / cos2theta

Hence Proved.

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