Question # If ( frac{cos x}{cos y}=n, frac{sin x}{sin y}=m, ) then what is ( left(m^{2}-n^{2}right) sin ^{2} y ) equal to?

# If ( frac{cos x}{cos y}=n, frac{sin x}{sin y}=m, ) then what is ( left(m^{2}-n^{2}right) sin ^{2} y ) equal to?

(a) ( 1-n^{2} )

(b)1 + n^{2}

(c) ( m^{2} )

(d) ( n^{2} )

Solution

( frac{cos x}{cos y}=n, frac{sin x}{sin y}=m )

( cos x=n cos y ; sin x=m sin y )

( sin ^{2} x+cos ^{2} x=1 )

( =sin ^{2} x^{2} sin ^{2} y+n^{2} cos ^{2} y=1 )

( Rightarrow m^{2} sin ^{2} y+n^{2}left(1-sin ^{2} yright)=1 )

( =1left(m^{2} sin ^{2} x+n^{2}-n^{2} sin ^{2} y=1right. )

( Rightarrowleft(m^{2}-n^{2}right) sin ^{2} y=1-n^{2} )

option (a)