Question

please solve this fast and show step by step​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1.

${ \cos}^{2} \theta \: - { \sin}^{2} \theta \: = \cos 2 \theta$

2.

$\displaystyle \: \: { \tan}^{2} \alpha = \frac{1 - \cos 2 \alpha }{1 + \cos2 \alpha }$

GIVEN

${ \tan}^{2} \alpha = { \cos}^{2} \phi \: - { \sin}^{2} \phi$

TO PROVE

${ \cos}^{2} \alpha \: - { \sin}^{2} \alpha = { \tan}^{2} \phi$

PROOF

It is given that

${ \tan}^{2} \alpha = { \cos}^{2} \phi \: - { \sin}^{2} \phi$

$\implies \: { \tan}^{2} \alpha = { \cos} \: 2 \phi \:$

So

${ \cos}^{2} \alpha \: - { \sin}^{2} \alpha$

$= { \cos} \: 2 \alpha \:$

$\displaystyle \: \: = \frac{1 - { \tan}^{2} \alpha }{1 + { \tan}^{2} \alpha }$

$\displaystyle \: \: = \frac{1 - { \cos} \: 2 \phi \: }{1 + { \cos} \: 2 \phi \: }$

$= { \tan}^{2} \phi$