Krishna
0

Step 1:  Explore the given question

            NOTE:  \cot\ A=\frac{7}{8}

                        \cot A=\frac{adjacent}{opposite}

                    Therefore, opposite side = 8

                                      Adjacent side = 7

                            

Step 2: Find the unknown side of the right triangle.

                

                 hypotenuse^2 = opposite^2 + adjacent^2

                hyp^2=8^2+7^2

                hyp=\sqrt{64+49}

                      hypotenuse = \sqrt{113}


Step 3:  Recall the formulas and find the \sin A , \cos A

            FRMULAS: (i) \sin A =\frac{opp}{hyp}

                                    \sin A=\frac{8}{\sqrt{113}}


                                (ii) \cos A =\frac{adj}{hyp}

                                    \cos A=\frac{7}{\sqrt{113}}


Step 4: Substitute the calculated ratio values in the given equation.

                  =   \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1+ \cos \theta)(1 - \cos \theta)}

                                         =   \frac{(1 )^2 - (\sin \theta)^2}{(1)^2 - (\cos \theta)^2} ( since (a + b)(a - b) = a^2 - b^2)

                                                          =      \frac{(1 )^2 - (\frac{8}{\sqrt{113}})^2}{(1)^2 - (\frac{7}{\sqrt{113}})^2}

                                                                                  =   \frac{\left(\frac{113-64}{113}\right)}{\left(\frac{113\ -\ 49}{133}\right)}

                                                                                                 =   \frac{49}{64}