Question

If Sec theta + tan theta=p,then find the value of sin theta in terms of p.

Answer 1

We know, sec²A - tan²A = 1
a² - b² = ( a + b )( a - b )


sec²A - tan²A = 1

( secA - tanA )( secA + tanA ) = 1



$\textbf{Given in the question, } secA + tanA = p$ -------1


( secA - tanA ) p = 1

secA - tanA = 1 / p. ----- 2




Adding 1 & 2 ,


secA + tanA + secA - tanA = p + 1 / p

2 secA = ( p² + 1 ) / p

secA = ( p² + 1 ) / 2p

1 / cosA = ( p² + 1 ) / 2p

cosA = 2p / ( p² + 1 )



Square on both sides,


cos²A = 4p² / ( p² + 1 )²


Multiply by - 1 on both sides,


- cos²A = - 4p² / ( p² + 1 )²


adding 1 on both sides,


1 - cos²A = 1 - 4p² / ( p² + 1 )²


we know, 1 - cos²A = sin²A


sin²A = [ p⁴ + 1 + 2p² - 4p² ] / ( p² + 1 )²

sin²A = [ ( p⁴ + 1 - 2p² ) / ( p² + 1 )²

sin²A = ( p² - 1 )² / ( p² + 1 )²

sinA = ( p² - 1 )( p² + 1 )


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