Question

if tan A+sin A = m and tan A - sin A=n then

(m^2-n^2)^2/mn

Answer 1

m =tanA + sinA

n =tanA - sinA

$m^{2}-n^{2} =(tanA + sinA)^{2} - (tanA-sinA)^{2}$

$m^{2}-n^{2} =4tanA.sinA$

$mn = (tanA+sinA)(tanA-sinA)$

$mn = tan^{2}A-sin^{2}A$

$mn = sin^{2}A(\frac{1}{cos^{2}A}-1)$

$mn = sin^{2}A(\frac{1-cos^{2}A}{cos^{2}A})$

$mn = sin^{2}A(\frac{sin^{2}A}{cos^{2}A})$

$mn = sin^{2}A.tan^{2}A$

$m^2- n^2 =4 sin A tan A$

$m^{2}-n^{2} =4\sqrt{mn}$