Question

if x=asec theta + btan theta and y = atan theta +bsec theta. prove that x square-y square=a square-b square

Answer 1

Answer:  The proof is done below.

Step-by-step explanation:  We are given that :

$x=a\sec\theta+b\tan\theta,\\\\y=a\tan\theta+b\sec\theta.$

We are to prove the following trigonometric equality :

$x^2-y^2=a^2-b^2.$

We will be using the following formulas :

$(i)~(a+b)^2=a^2+2ab+b^2,\\\\(ii)~\sec^2\theta-\tan^2\theta=1.$

The proof of the given equality is as follows :

$L.H.S.\\\\=x^2-y^2\\\\=(a\sec\theta+b\tan\theta)^2-(a\tan\theta+b\sec\theta)^2\\\\=(a^2\sec^2\theta+2ab\sec\theta\tan\theta+b^2\tan^2\theta)-(a^2\tan^2\theta+2ab\sec\theta\tan\theta+b^2\sec^2\theta)\\\\=a^2(\sec^2\theta-\tan^2\theta)-b^2(\sec^2\theta-\tan^2\theta)\\\\=a^2\times1-b^2\times 1\\\\=a^2-b^2\\\\=R.H.S.$

Thus, we get

$x^2-y^2=a^2-b^2.$

Hence proved.

Answer 2

Answer:

therefore

x square - y square = a square - b square