Question

Prove √(Sec theta-1/sec theta+1)=cosec theta-cot theta

Answer 1

lets replace theta with ' A ' and under root( x  )by sqrt(x) LHS becomes:

sqrt((sec(A)-1)/(sec(A)+1)) + sqrt((sec(A)+1)/(sec(A)-1))
=
sec(A)-1 + sec(A)+1/ sqrt((sec(A)+1)*(sec(A)-1))   : fraction adding
=
2sec(A)/sqrt(sec^2(A)-1)           :sec square A -> sec^2(A) 
=
2sec(A)/sqrt((tan^2(A))
=
2sec(A)/tan(A)
=
2cosec(A)

Answer 2

Hi ,

Here I used A instead of theta.

LHS = √( secA - 1) / ( secA + 1 )

=√[(secA-1)²/(secA+1)(secA+1)

= √(secA -1 )²/(sec² A - 1 )

= √(secA - 1 )²/( tan² A )

= ( SecA - 1 ) / tanA

= SecA / tanA - 1/tanA

= ( 1/cosA ) / ( sinA/cosA ) - cotA

= 1/sinA - cotA

= CosecA - cotA

= RHS

I hope this helps you.

:)