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

Question

Philoid · Accepted Answer

Given: tan A = n tan B

Therefore, tan B = tan A/n

Thus, cot B = n/tan A

⇒ 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 =

⇒ 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.