Question

If tanΦ+sinΦ=m and tanΦ-sinΦ show that m^2-n^2 = 4√(mn). Please answer neatly and explain completely

Answer 1

Here is the solution in the below pics:-




Hope it helps u..

Answer 2

tan∅ + sin∅ = m--------(1)
tan∅ - tan∅ = n -------(2)

take square both equations and subtract (1) - (2)

( tan∅+ sin∅)² - ( tan∅-sin∅)² = m²-n²

tan²∅ +sin²∅+2sin∅.tan∅ -tan²∅-sin²∅+2sin∅.tan∅ = m²-n²

4tan∅.sin∅ = m² - n² ---------(2)

now,

multiply equation (1)and (2)
(tan∅+ sin∅)(tan∅-sin∅) = mn
tan²∅-sin²∅ =mn

sin²∅/cos²∅ - sin²∅ = mn

sin²∅( 1 - cos²∅)/cos²∅ = mn
sin²∅(sin²∅)/cos²∅ = mn
sin²∅.tan²∅ =mn
sin∅.tan∅ =√mn

now , value of sin∅.tan∅ put in eqn (2)

4√mn = m² -n²

hence, proved//