Question

Show that ( SecA - TanA)² = 1-SinA/1 +SinA

Answer 1

To prove --->

( SecA - tanA )² = ( 1 - SinA ) / ( 1 + SinA )

Proof--->

LHS = ( SecA - tanA )²

We know that ,

SecA = 1 / CosA and tanA = SinA /CosA , applying it here we get,

= { ( 1 / CosA ) - ( SinA / CosA ) }²

Taking LCM as CosA , we get,

= { ( 1 - SinA ) / CosA }²

= ( 1 - SinA )² / Cos²A

We know that, Cos²A = 1 - Sin²A ,we get,

= ( 1 - SinA )² / ( 1 - Sin²A )

= ( 1 - SinA )² / ( 1 )² - ( SinA )²

Applying a² - b² = ( a + b ) ( a - b ) , we get,

= ( 1 - SinA )² / ( 1 + SinA ) ( 1 - SinA )

( 1 - SinA ) is cancel out from numerator and denominator, we get,

= ( 1 - SinA ) / ( 1 + SinA ) = RHS

Additional information--->

1) 1 + tan²A = Sec²A

2) 1 + Cot²A = Cosec²A