Question

How do you simplify the expression $\dfrac{{\sin x - \cos x}}{{\sin x\cos x}}$

Answer 1

Hint : In the question we are provided with a trigonometric expression and we would need to simplify it using the trigonometric properties. Firstly, spit the numerator into two terms and then cancel the common quantities and then use the common trigonometric formulas.
Formula used:
$\dfrac{1}{{\cos x}} = \sec x$ and $\dfrac{1}{{\sin x}} = \cos ec x$

Complete step-by-step answer :
We need to simplify this trigonometric expression. The trigonometric expression is defined as the quantities which contain trigonometric ratios.
The given expression is $\dfrac{{\sin x - \cos x}}{{\sin x\cos x}}$
Splitting the numerator $\dfrac{{\sin x}}{{\sin x\cos x}} - \dfrac{{\cos x}}{{\sin x\cos x}}$
Now, if we noticed that $\sin x$ can be cancelled from the first term and the $\cos x$ would get cancelled from the second term.
Cancelling the common terms,
$\dfrac{1}{{\cos x}} - \dfrac{1}{{\sin x}}$
Now, these quantities have their respective formula such that $\dfrac{1}{{\cos x}} = \sec x$ and $\dfrac{1}{{\sin x}} = \cos ec x$
Replacing the terms with these formulas,
$\sec x - \cos ec x$
So, the required answer or simplification of the given expression is
$\dfrac{{\sin x - \cos x}}{{\sin x\cos x}}$ = $\sec x - \cos ec x$
So, the correct answer is “$\sec x - \cos ec x$”.

Note : All the trigonometric properties should be learnt on tips. You don’t need to open the formula sheet for solving such basic questions. The trigonometric functions are very easy to memorize, you just need to practice a lot.
Here we use the given formulae:
$\dfrac{1}{{\cos x}} = \sec x$ and $\dfrac{1}{{\sin x}} = \cos ec x$