Question

How do you simplify the expression \[\dfrac{{\sin x + \cos x}}{{\sin x\cos x}}\]?

Answer 1

Hint: Here, we will split the denominator and then simplify it by canceling the common terms. We will use reciprocal trigonometric ratios to simplify the given expression. The trigonometric expression is defined as an expression involving the trigonometric ratios. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle.

Formula Used:
We will use following formulas:
1. \[\sec x = \dfrac{1}{{\cos x}}\]
2. \[\csc x = \dfrac{1}{{\sin x}}\]

Complete step-by-step solution:
We will use trigonometric ratios and identities to simplify the given expression.
Simplifying the given expression by splitting the denominator, we get
\[\dfrac{{\sin x + \cos x}}{{\sin x\cos x}} = \dfrac{{\sin x}}{{\sin x\cos x}} + \dfrac{{\cos x}}{{\sin x\cos x}}\]
The right hand side is the sum of two fractions.
The numerator and denominator of the first fraction have the common factor \[\sin x\].
The numerator and denominator of the second fraction have the common factor \[\cos x\].
Factoring the common factors, we get
\[ \Rightarrow \dfrac{{\sin x + \cos x}}{{\sin x\cos x}} = \dfrac{1}{{\cos x}} \cdot \dfrac{{\sin x}}{{\sin x}} + \dfrac{1}{{\sin x}} \cdot \dfrac{{\cos x}}{{\cos x}}\].
Simplifying the expression, we get
\[\begin{array}{l} \Rightarrow \dfrac{{\sin x + \cos x}}{{\sin x\cos x}} = \dfrac{1}{{\cos x}} \cdot 1 + \dfrac{1}{{\sin x}} \cdot 1\\ \Rightarrow \dfrac{{\sin x + \cos x}}{{\sin x\cos x}} = \dfrac{1}{{\cos x}} + \dfrac{1}{{\sin x}}\end{array}\].
Now, we will rewrite the right-hand side expression using the trigonometric ratios.
The secant of an angle \[x\] can be written as the reciprocal of cosine of the angle \[x\]. This can be written as \[\sec x = \dfrac{1}{{\cos x}}\].
The cosecant of an angle \[x\] can be written as the reciprocal of sine of the angle \[x\]. This can be written as \[\csc x = \dfrac{1}{{\sin x}}\].
Substituting \[\dfrac{1}{{\cos x}} = \sec x\] and \[\dfrac{1}{{\sin x}} = \csc x\] in the expression, we get
\[\therefore \dfrac{\sin x+\cos x}{\sin x\cos x}=\sec x+\csc x\]

Therefore, the required value of the given expression \[\dfrac{{\sin x + \cos x}}{{\sin x\cos x}}\] is \[\sec x + \csc x\].

Note:
We used the trigonometric ratios of sine, cosine, secant, and cosecant in the solution.
The sine of an angle of a right-angled triangle is the ratio of its perpendicular and hypotenuse. This can be written as \[\sin x = \dfrac{P}{H}\].
The cosine of an angle of a right-angled triangle is the ratio of its base and hypotenuse. This can be written as \[\cos x = \dfrac{B}{H}\].
The secant of an angle of a right-angled triangle is the ratio of its hypotenuse and base. This can be written as \[\sec x = \dfrac{H}{B}\].
The cosecant of an angle of a right-angled triangle is the ratio of its hypotenuse and perpendicular. This can be written as \[\csc x = \dfrac{H}{P}\].