Question

If tan A = ntan B and sin A=m sin B ,prove that cos^2A =m^2-1/n^2-1

Answer 1

Answer:

Step-by-step explanation:

SinA = m Sin B ,

Sin A/ Sin B = m and

tan A= n tan B,

tan A/tan B= n or

Sin A/ cos A/Sin B/cos B= n ,

SinA Cos B / Cos A. Sin B= n

m2 -1/ n2 -1 = (Sin A / Sin B )2 -1/(Sin ACos B /Cos A Sin B)2 -1

= Sin2A/ Sin2B -1/ Sin2ACos2B/Cos2ASin2B -1

= Sin2A-Sin2B/ Sin2B/Sin2ACos2B - Cos2A Sin2B/ Cos2A Sin2B

= Sin2A - Sin2B x Cos2A Sin2B/Sin2B (Sin2A Cos2B - Cos2A Sin2B)

= Sin2A - Sin2B x Cos2A/ Sin2A(1- sin2B) - (1-Sin2A) Sin2B

= (Sin2A - Sin 2B ) x Cos2A /Sin2A - Sin2A Sin2B - Sin2B +Sin2A Sin2B

= (Sin2A - Sin2B) x Cos2A/ ( Sin2A - Sin2B)

= Cos 2A