Question

Solve the trigonometric expression $\sqrt {\dfrac{{\sec \theta - 1}}{{\sec \theta + 1}}} + \sqrt {\dfrac{{\sec \theta + 1}}{{\sec \theta - 1}}} = ?$

Answer 1

Hint: According to given in the question we have to solve the given trigonometric expression so, first of all to solve the given trigonometric expression we have to convert $\sec \theta $in the form of $\cos \theta $with the help of the formula as given below:
$\sec \theta = \dfrac{1}{{\cos \theta }}$………………….(1)
Now, after using the formula (1) we will get the whole given trigonometric expression in the form of $\cos \theta $ Now to solve the obtained expression we have to take the L.C.M. to obtain the expression in easy form. Now, we have to multiply the obtained denominator with the numerator and denominator to solve the expression obtained.
So, we will get the expression in the form of $\sin \theta $and to solve further we have to convert $\cos ec\theta $ in the form of ${\sin ^2}\theta $ with the help of the formula as given below:

Formula used:
$\sin \theta = \dfrac{1}{{\cos ec\theta }}$………………………..(2)

Complete step by step answer:
Step 1: As given in the question to solve the trigonometric expression first of all we have to convert $\sec \theta $in the form of $\cos \theta $with the help of the formula (1) as mentioned in the solution hint.
$ = \sqrt {\dfrac{{\dfrac{1}{{\cos \theta }} - 1}}{{\dfrac{1}{{\cos \theta }} + 1}}} + \sqrt {\dfrac{{\dfrac{1}{{\cos \theta }} + 1}}{{\dfrac{1}{{\cos \theta }} - 1}}} $
Step 2: Now, to solve the trigonometric expression just obtained above we have to take L.C.M.
$ = \sqrt {\dfrac{{\dfrac{{1 - \cos \theta }}{{\cos \theta }}}}{{\dfrac{{1 + \cos \theta }}{{\cos \theta }}}}} + \sqrt {\dfrac{{\dfrac{{1 + \cos \theta }}{{\cos \theta }}}}{{\dfrac{{1 - \cos \theta }}{{\cos \theta }}}}} $
On eliminating the $\cos \theta $from the obtained expression,
$\sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }}} + \sqrt {\dfrac{{1 + \cos \theta }}{{1 - \cos \theta }}} $
Step 3: Now, we have to multiply the obtained denominator with the nominator and denominator to solve the expression obtained.
$ = \sqrt {\dfrac{{1 - \cos \theta }}{{1 + \cos \theta }} \times \dfrac{{1 - \cos \theta }}{{1 - \cos \theta }}} + \sqrt {\dfrac{{1 + \cos \theta }}{{1 - \cos \theta }} \times \dfrac{{1 + \cos \theta }}{{1 + \cos \theta }}} $
$ = \sqrt {\dfrac{{{{(1 - \cos \theta )}^2}}}{{1 - {{\cos }^2}\theta }}} + \sqrt {\dfrac{{{{(1 + \cos \theta )}^2}}}{{1 - {{\cos }^2}\theta }}} $
Step 4: On solving the square root, and using the formula (2) as mentioned in the solution hint.
$
   = \dfrac{{1 - \cos \theta }}{{\sin \theta }} + \dfrac{{1 + \cos \theta }}{{\sin \theta }} \\
   = \dfrac{{1 - \cos \theta + 1 + \cos \theta }}{{\sin \theta }} \\
 $
Step 5: On eliminating $\cos \theta $ from the expression obtained just above.
$\dfrac{2}{{\sin \theta }}$
With the help of the formula as mentioned in the solution hint,
$2\cos ec\theta $

Hence, with the help of the formulas as mentioned in the solution hint we have obtained the solution of the given trigonometric expression$\sqrt {\dfrac{{\sec \theta - 1}}{{\sec \theta + 1}}} + \sqrt {\dfrac{{\sec \theta + 1}}{{\sec \theta - 1}}} = 2\cos ec\theta $

Note:
It is necessary to convert $\sec \theta $in the form of $\cos \theta $to make the calculations easy for the given trigonometric expression.
In the sublimation method we have to multiply the denominator with the numerator and denominator to solve the expression.