Question

Simplify the expression $\dfrac{\cos x}{\sin x}$

Answer 1

The trigonometric function sinx is the ratio of the length of the side opposite to angle x and length of the hypotenuse and the function cosx is the ratio of the length of the side adjacent to angle x and length of the hypotenuse. The expression $\dfrac{\cos x}{\sin x}$ gives the ratio of the length of the side adjacent to angle x and length of the side opposite to angle x which is equal to a trigonometric function which we need to find.

Complete step-by-step solution:
Trigonometric functions are the real functions that relate the angle in a right-angled triangle to the ratio of its length.
Sine is the trigonometric function of any specified angle that is used in the context of a right angle.
It is usually defined as the ratio of the length of the side opposite to an angle to the length of the hypotenuse of the right-angle triangle.
sinx= $\dfrac{\text{length of the side opposite to angle x}}{\text{ the length of the hypotenuse}}$.
Cos is the trigonometric function of any specified angle that is used in the context of a right angle.
It is usually defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse of the right-angle triangle.
cosx= $\dfrac{\text{length of the side adjacent to angle x}}{\text{the length of the hypotenuse}}$.
In the given question,
we need to simplify the expression $\dfrac{\cos x}{\sin x}$
Upon substituting the formulae of the sine and the cosine, we get,
$\dfrac{\cos x}{\sin x}$ = $\dfrac{\dfrac{\text{length of the side adjacent to angle x }}{\text{ the length of the hypotenuse)}}} {\dfrac{\text{length of the side opposite to angle x}}{\text{ the length of the hypotenuse}}}$
From the above,
cotx = $\dfrac{\text{length of the side adjacent to angle x}}{\text{ the length of the side opposite to angle x}}$
Substituting the same,
We get $\dfrac{\cos x}{\sin x}=\cot x$
Cotangent is the trigonometric function of any specified angle that is used in the context of a right angle just like any trigonometric function such as sine, cosine, etc.
The expression $\dfrac{\cos x}{\sin x}$ on simplification is equal to trigonometric function cotx.

Note: We need to know the basic formulae of all the trigonometric functions to solve the expression given numerically. The value of the expression can be cross checked by giving a value to angle x.
$\dfrac{\cos x}{\sin x}=\cot x$
Taking the value of angle x equal to 45.
LHS:
$\Rightarrow \dfrac{\cos {{45}^{\circ }}}{\sin {{45}^{\circ }}}=\cot x$
$\Rightarrow \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{2}}}=\cot x$
Which is equal to 1.
RHS:
$\Rightarrow \cot x=\cot {{45}^{\circ }}$
$\Rightarrow cotx=1$
LHS = RHS
Hence, the result attained is correct.