Question

Prove the following trigonometric identities

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x² − y² = a² − b²

Answer 1

`Given that

`x = a sec theta + b tan theta`

`y = a ta theta +  b sec theta`

We have to prove  `x^2 - y^2 = a^2 - b^2`

We know that `sec^2 theta - tan^2 theta  = 1`

So,

`x^2 - y^2`

`= (a sec theta + b tan theta)^2 - (a tan theta + b sec theta)^2`

`= (a^2 sec^2 theta + 2 ab sec theta + b^2 tan^2 theta) - (a^2 tan^2 theta +  2 ab sec theta tan theta + b^2 + sec^2 theta)`

`= a^2 (sec^2 theta  -  tan^2 theta) - b^2 (sec^2 theta -  tan^2 theta)`

`= a^2 - b^2 `

Hence proved.