If tan theta sin theta m and tan theta sin theta n then?

Answer: Given,

tan⁡θ+sinθ ⁡= m..(1) 

tan⁡θ – sin⁡θ = n…(2)

Squaring both sides and subtracting equation (2) from equation (1).m²–n²=tan²⁡θ⁡+sin²θ⁡ ⁡+ 2tan⁡θ⁡sin⁡θ⁡–tan²⁡θ⁡+sin²θ⁡ -2tanθ⁡⁡sinθ⁡⁡

=4tanθ⁡sin⁡θ (i) 

mn=(tan⁡+sin⁡)(tan⁡-sin⁡)

mn=tan2θ⁡ – sin²θ⁡

mn=sin2⁡θ / cos²θ ⁡– sin²θ⁡

mn=sin²θ⁡sec²θ-1

mn=sin²⁡θtan²θ

Taking square root on both sides,

√mn=sin⁡θtanθ⁡..(3)  

From (1) and (3),

m²–n²=4√mn .