Question

If sectheta - tan theta =root 2tan theta then show that sectheta +tan theta =root 2sec theta

Answer 1

secΘ-tanΘ=√2 tanΘ        {Given}
On squaring,
(secΘ-tanΘ)^2=(√2 tanΘ)^2        
sec^2Θ+tan^2Θ-2secΘtanΘ=2tan^2Θ
-2secΘtanΘ=2tan^2Θ-sec^2Θ-tan^2Θ
-2secΘtanΘ=tan^2Θ-sec^2Θ
-2secΘtanΘ= -1                   {Because 1+tan^2Θ=sec^2Θ}
2secΘtanΘ= 1                →1
To prove secΘ+tanΘ=√2 secΘ
LHS 
secΘ+tanΘ       
On squaring,  
(secΘ+tanΘ)^2
sec^2Θ+tan^2Θ+2tanΘsecΘ
sec^2Θ+tan^2Θ+1               {Because 1+tan^2Θ=sec^2Θ}
sec^2Θ+sec^2Θ
2sec^2Θ
On rooting,
√2 secΘ
Therefore secΘ+tanΘ=√2 secΘ