Question

Answer the above question ​

Answer 1

hope this answer will help you.

Answer 2

tan  θ + sin  θ = m ......( 1 )

tan  θ - sin  θ = n .......( 2 )

Multiply eq (1) and eq (2) :

( tan θ + sin θ )( tan θ - sin θ ) = m n

⇒ tan² θ - sin² θ = mn

⇒ sin² θ / cos² θ - sin² θ = mn

⇒ sin² θ ( 1/cos² θ - 1 ) = mn

⇒ sin² θ ( 1 - cos² θ ) / cos² θ = mn

⇒ sin² θ/cos² θ × sin² θ = mn

⇒ sin² θ × tan² θ = mn

Multiply both sides by 16 :

⇒ 16 sin² θ tan² θ = 16 mn

⇒ ( 4 sin  θ tan  θ )² = 16 mn

Add and subtract these terms :

⇒ ( sin² θ + tan² θ + 2 sin θ tan θ + 2 sin  θ tan  θ - sin² θ - tan² θ )² = 16 mn

Use a² + b² + 2 ab as ( a + b )²

⇒ ( ( sin θ + tan θ )² - ( sin θ - tan θ )² )² = 16 mn

Recall the values of m and n :

⇒ ( m² - n² )² = 16 mn

Hence the given problem is proved :)