Question

If sectetha - tantetha is p then show that costheta is 2p/ psquare + one

Answer 1

Let theta = alpha for the sake of my typing comfort,

Let equation 1 be:
$\sec \alpha - \tan\alpha = p \\ \:$
As we know,
$1 + { \tan}^{2} \alpha = { \sec }^{2} \alpha \\ { \sec }^{2} \alpha - { \tan}^{2} \alpha = 1 \\ ( \sec \alpha - \tan \alpha )( \sec \alpha + \tan \alpha ) = 1$
substituting the values,

and equation 2 be:
$\sec \alpha + \cot \alpha = \frac{1}{p}$

adding eq. 1 and 2
$2 \sec \alpha = \frac{1}{p} + p$
Which in furthur solving gives,
$2 \sec \alpha = \frac{1 + {p}^{2} }{p} \\ \sec \alpha = \frac{1 + {p}^{2} }{2p } \\ \cos\alpha = \frac{2p}{1 + {p}^{2} }$
Hence proved.