Question

If tanA=n tanB and sin A =m sin B, prove that n2 - 1.

Answer 1

Correct question :

• If TanA = n Tan B and Sin A = m Sin B , prove that cos² A = m² - 1 / n² - 1.

Proof :

Tan A = n Tan B

Tan B = Tan A / n

Taking the reciprocal

1 / Tan B = n / Tan A

Cot A = n / Tan A _____(1)

Sin A = m Sin B

Sin B = Sin A / m

Taking the reciprocal

1/ Sin B = m / Sin A

Cosec B = m / sin A ____(2)

Using the identity

• cosec² B = 1 + cot² B
• Cosec ² B - cot² B = 1

Putting value of Cosec B and Cot B from equation 2 and 1 respectively

(m / sin A)² - ( n / Tan A )² = 1

m² / Sin² A - n² / Tan ² A = 1

Tan² A = sin² A / cos² A

→ 1/ Tan² A = cos² A / Sin² A

m² / sin² A - n² cos² A / sin² A = 1

$\tt \dfrac{ {m}^{2} - {n}^{2} {cos}^{2} A }{ {sin}^{2} A} = 1$

m² - n² cos² A = 1 × sin² A

m ² - n² cos² A = sin² A

Using identity

• sin² A + cos² A = 1
• Sin² A = 1 - cos² A

m² - n² cos² A = 1 - cos² A

Rearranging the terms :

m² - 1 = n² cos² A - cos² A

m² - 1 = cos² A ( n² - 1 )

m² - 1 / n² - 1 = cos² A

Or

Cos² A = m² - 1 / n² - 1

Hence proved