Question

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question number 19...

if A, B and C....in attatchment..

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Answer 1

Use the triangle's angle sum property.

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Aryan Jha

Answer 2

If A , B , C are Interior Angles of Triangle ABC, Then :

✿  Sum of Angles A , B , C of the Triangle ABC should be Equal to 180°

⇒ A + B + C = 180°

⇒ B + C = 180° - A

$\mathsf{Given : Cosec^2(\frac{B + C}{2}) - Tan^2(\frac{A}{2})}$

$\mathsf{\implies Cosec^2(\frac{180 - A}{2}) - Tan^2(\frac{A}{2})}$

$\mathsf{\implies Cosec^2(\frac{180}{2} - \frac{A}{2} ) - Tan^2(\frac{A}{2})}$

$\mathsf{\implies Cosec^2(90 - \frac{A}{2} ) - Tan^2(\frac{A}{2})}$

$\mathsf{We\;know\;that : Cosec(90 - \theta) = Sec\theta}\\\\\mathsf{\implies Cosec^2(90 - \theta) = Sec^2\theta}\\\\\mathsf{\implies Cosec^2(90 - \frac{A}{2}) = Sec^2(\frac{A}{2})}$

$\mathsf{\implies Sec^2(\frac{A}{2}) - Tan^2(\frac{A}{2})}$

$\mathsf{We\;know\;that : Sec^2\theta - Tan^2\theta = 1}$

$\mathsf{\implies Sec^2(\frac{A}{2}) - Tan^2(\frac{A}{2}) = 1}$