# If tan A = n tan B and sin A = m sin B, prove that .

Given: tan A = n tan B

Therefore,

Thus,

Squaring both sides, we get,

cot2 B = n2/tan2 A ……(1)

Also, sin A = m sin B

Therefore, sin B = sin A/m

Thus, cosec B = m/sin A

cosec2 B = m2/sin2 A ……(2)

Now, subtract equation (2) from (1):

cosec2 B – cot2 B =

1 =

1 =

m2 – n2 cos2 A = sin2 A

m2 – n2 cos2 A = 1 – cos2 A

m2 – 1 = n2 cos2 A – cos2 A

(n2 – 1)cos2 A = m2 – 1

cos2 A = (m2 – 1)/(n2 – 1)

Hence, proved.

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