Question

How do you simplify $\dfrac{{1 + \sin y}}{{1 + \csc y}}$ ?

Answer 1

Hint : The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\csc (x) = \dfrac{1}{{\sin (x)}}$ . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step-by-step answer :
In the given problem, we have to simplify the ratio given to us in the question, $\dfrac{{1 + \sin y}}{{1 + \csc y}}$ .
So, $\dfrac{{1 + \sin y}}{{1 + \csc y}}$
Using $\csc \theta = \dfrac{1}{{\sin \theta }}$ , we get,
$\Rightarrow$ $\dfrac{{1 + \sin y}}{{1 + \dfrac{1}{{\sin y}}}}$
Now, taking LCM in the denominator, we get,
$\Rightarrow$ $\dfrac{{1 + \sin y}}{{\dfrac{{\sin y + 1}}{{\sin y}}}}$
Simplifying the expression further, we get,
$\Rightarrow$ $\dfrac{{\left( {1 + \sin y} \right)\sin y}}{{\left( {1 + \sin y} \right)}}$
Cancelling out the common factors in numerator and denominator, we get,
$\Rightarrow$ $\sin y$
Hence, the rational trigonometric function in y $\dfrac{{1 + \sin y}}{{1 + \csc y}}$ can be simplified as $\sin y$ by the use of basic algebraic rules and simple trigonometric formulae.
So, the correct answer is “$\sin y$ ”.

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.