Question

How do you simplify: $\left( {\dfrac{{1 + \sin u}}{{\cos u}}} \right) + \left( {\dfrac{{\cos u}}{{1 + \sin u}}} \right)$ ?

Answer 1

Hint : The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. If two fractions with different denominators are to be added, then we have to first take LCM of denominators without changing the value of fractions and then add up the numerators.

Complete step-by-step answer :
In the given problem, we have to simplify the sum of $\left( {\dfrac{{1 + \sin u}}{{\cos u}}} \right)$ and $\left( {\dfrac{{\cos u}}{{1 + \sin u}}} \right)$ .
So, we have to add up two rational trigonometric functions in x with different denominators. Hence, we need to take the LCM of denominators without changing the value of the fraction.
So, $\left( {\dfrac{{1 + \sin u}}{{\cos u}}} \right) + \left( {\dfrac{{\cos u}}{{1 + \sin u}}} \right)$
$\Rightarrow \left( {\dfrac{{1 + \sin u}}{{\cos u}}} \right)\left( {\dfrac{{1 + \sin u}}{{1 + \sin u}}} \right) + \left( {\dfrac{{\cos u}}{{1 + \sin u}}} \right)\left( {\dfrac{{\cos u}}{{\cos u}}} \right)$
$\Rightarrow \left( {\dfrac{{1 + {{\sin }^2}u + 2\sin u}}{{\cos u + \cos u\sin u}}} \right) + \left( {\dfrac{{{{\cos }^2}u}}{{\cos u + \cos u\sin u}}} \right)$
$\Rightarrow \left( {\dfrac{{1 + {{\sin }^2}u + 2\sin u + {{\cos }^2}u}}{{\cos u + \cos u\sin u}}} \right)$
Using trigonometric identity ${\sin ^2}x + {\cos ^2}x = 1$ , we get,
$\Rightarrow \left( {\dfrac{{2 + 2\sin u}}{{\cos u + \cos u\sin u}}} \right)$
$\Rightarrow \dfrac{{2\left( {1 + \sin u} \right)}}{{\cos u\left( {1 + \sin u} \right)}}$
Cancelling $\left( {1 + \sin u} \right)$ in numerator as well as denominator, we get,
$\Rightarrow \dfrac{2}{{\cos u}}$
We know the trigonometric formula $\sec \theta = \dfrac{1}{{\cos \theta }}$ . So, we get,
$\Rightarrow 2\sec u$
Hence, the sum $\left( {\dfrac{{1 + \sin u}}{{\cos u}}} \right) + \left( {\dfrac{{\cos u}}{{1 + \sin u}}} \right)$ can be simplified as $2\sec u$ by the use of basic algebraic rules and simple trigonometric formulae.
So, the correct answer is “$2\sec u$”.

Note : Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: $\tan (x) = \dfrac{{\sin (x)}}{{\cos (x)}}$ and $\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}}$ . Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such type of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers.