Krishna
0

Step 1: Take down the given equation

              GIVEN: \sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A


Step 2: Take the L.H.S of the equation and think about the possible ways to prove R.H.S.

                NOTE: \sqrt{\frac{1 + \sin A}{1 - \sin A}}


Step 3: Rationalize the irrational denominator of L.H.S

            EXPLANATION; You can't solve an equation that contains a fraction with an

            irrational denominator, which means that the denominator contains a term

            with a radical sign. This includes square, cube and higher roots. Getting rid

            of the radical sign is called rationalizing the denominator.


            EXAMPLE: \sqrt{\frac{1 + \sin A}{1 - \sin A}}

                                Divide and multiply by 1 + \sin A


                             =   \sqrt{\frac{1 + \sin A}{1 - \sin A} \frac{1 + \sin A}{1 + \sin A}}


                             =   \sqrt{\frac{(1 + \sin A )^2}{(1 - \sin A)(1 + \sin A)}}


                            =   \sqrt{\frac{(1 + \sin A)^2}{1^2 - \sin^2 A}}


                            =   \frac{1 + \sin A}{\cos^2 A}          (Since, \sin^2 \theta + \cos^2 \theta = 1 )


                        Separate the denominator and simplify for R.H.S


                            =   \frac{1}{\cos A} + \frac{ \sin A}{\cos A}


                            = \sec A + \tan A

                                              (since, \sec \theta = \frac{1}{\cos \theta}

                                                           \tan \theta = \frac{\sin \theta}{\cos \theta} )


                             Hence, proved