Question

If tan (teta) + sin (teta) =m and tan(teta) -sin (teta) = n , then show that (m^2 - n^2)^2 =16mn or (m^2-n^2)= 4 root mn.

Answer 1

m = tanФ + sinФ , n = tanФ - sinФ
therefore , m - n = 2sinФ
also m + n = 2tanФ
mn = tan²Ф- sin²Ф = sin²Ф(1 - cos²Ф) / cos²Ф
mn = sin^4Ф / cos²Ф = sin²Ф/cos²Ф * sin²Ф = tan²Ф * sin²Ф
therefore m² - n² = (m+n)(m-n) = 4tanФsinФ = 4 √(tan²Ф * sin²Ф)= 4√mn
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