Question

What is the formula of \[\dfrac{{(1 - \cos x)}}{{\sin x}}\]?

Answer 1

Hint: In this problem, we need to solve the trigonometric function by using Pythagorean trigonometric identity. The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity is used here to solve this trigonometric expression. In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. Here, we use the trigonometric identity is \[2{\sin ^2}(\dfrac{x}{2}) = 1 - \cos x\].

Complete step by step solution:
In the given problem,
\[\dfrac{{(1 - \cos x)}}{{\sin x}}\]
From the trigonometric identities,
\[2{\sin ^2}(\dfrac{x}{2}) = 1 - \cos x\]
\[\sin x = 2\sin (\dfrac{x}{2})\cos (\dfrac{x}{2})\]
Applying these trigonometric identities in the given function, we can get
\[\dfrac{{(1 - \cos x)}}{{\sin x}} = \dfrac{{2{{\sin }^2}(\dfrac{x}{2})}}{{2\sin \left( {\dfrac{x}{2}} \right)\cos \left( {\dfrac{x}{2}} \right)}}\]
By simplifying, we can get
\[ = \dfrac{{\sin (\dfrac{x}{2})}}{{\cos (\dfrac{x}{2})}}\]
On comparing this formula,\[\dfrac{{\sin x}}{{\cos x}} = \tan x\], we can get
\[\dfrac{{(1 - \cos x)}}{{\sin x}} = \tan \left( {\dfrac{x}{2}} \right)\]
Therefore, the formula of \[\dfrac{{(1 - \cos x)}}{{\sin x}}\]is\[\tan (\dfrac{x}{2})\].

Additional information: sin (x) is an odd function because sin(-x) = -sin(x). It's graph is symmetric to the origin. Cos (x) on the other hand is an even function cos (-x) = cos (x), and its graph is symmetric to the y-axis. There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are abbreviated as sin, cos, tan, cosec, sec, cot.

Note:
Here, we can use Pythagorean trigonometric identity by solving this function.Trigonometric identities are trigonometry equations that are always true, and they're often used to solve trigonometry and geometry problems and understand various mathematical properties. Knowing key trig identities helps you remember and understand important mathematical principles and solve numerous math problems.