Question

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Answer 1

Solution -

Given:

SinA = mSinB 

m/SinA = 1/SinB

As 1/Sinθ = Cosecθ

Therefore, CosecB = m/SinA --------------1

Similarly,

TanA = nTanB

1/TanB = n/TanA

As Cotθ = 1/Tanθ

Therefore,  CotB = n/TanA  ---------------2

We know that,

Cosec²θ - Cot²θ = 1

Hence, Cosec²B - Cot²B = 1

Substitute the value of CosecB and CotB from equation 1 and equation 2

(m/SinA)² - (n/TanA)² = 1

m²/Sin²A - n² Cos²A/Sin²A = 1  

(As Tan^2A = Sin^2A/Cos^2A)

m² - n² Cos²A = Sin²A

m² - n² Cos²A = 1 - Cos²A        

 (Sin²A = 1 - Cos²A)

n²Cos²A - Cos²A = m² - 1

Cos²A (n² - 1) = m² - 1

Cos²A = (m² - 1) / (n² - 1)

Hence proved.