PROVE THAT cos 2 tita / 1 + sin 2 tita = tan ( pi/4 - tita ) - Maths - Trigonometric Functions

Question

PROVE THAT cos 2 tita / 1 + sin 2 tita = tan ( pi/4 - tita )

Ankush Jain · Accepted Answer

To prove: $\frac{\cos 2 θ}{1 + \sin 2 θ} = \tan (\frac{π}{4} - θ) o r \frac{1 - \tan θ}{1 + \tan θ}$ Proof: LHS = $\frac{\cos 2 θ}{1 + \sin 2 θ} = \frac{\frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}}{1 + \frac{2 \tan θ}{1 + \tan^{2} θ}}$ [Using: sin 2x =

and cos 2x =

] = $\frac{\frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}}{\frac{1 + \tan^{2} θ + 2 \tan θ}{1 + \tan^{2} θ}} = \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ + 2 \tan θ}$ = $\frac{(1 - \tan θ) (1 + \tan θ)}{{(1 + \tan θ)}^{2}} = \frac{(1 - \tan θ)}{(1 + \tan θ)}$ = $\frac{\tan \frac{π}{4} - \tan θ}{\tan \frac{π}{4} + \tan θ} = \tan (\frac{π}{4} - θ)$ = RHS [Hence proved]