Question

If Cos α / Cos β = m and Cos α / Sin β = n then prove that (m² + n²) Cos² β = n².

Answer 1

GIVEN:
Cos α / Cos β = m  ……………(1)
Cos α / Sin β = n ……………….(2)

LHS = (m² + n²) Cos² β
= [(Cos α / Cos β)² + (Cos α / Sin β)]Cos² β

[From eq 1 & 2]

=  [(Cos² α / Cos² β)² + (Cos² α / Sin² β)]Cos² β
= [(Cos² α Sin² β + Cos² α Cos² β)/ Cos² βSin² β] Cos² β

= Cos² α[(Sin² β + Cos² β)/Cos² βSin² β] Cos² β

= Cos² α[(1) / Cos² βSin² β] Cos² β

[ sin² θ + cos²θ = 1]

LHS = Cos² α / Sin² β
LHS = (Cos α / Sin β)²
LHS = n²

LHS = RHS

Hence, proved

HOPE THIS WILL HELP YOU...

Answer 2

the answer is in the attachment