Question

How do you simplify \[{\csc ^2}\phi - 1\]?

Answer 1

Hint: In the above question, the concept is based on the concept of trigonometry. The main approach towards solving the above expression is by using trigonometric identities. The expression has cosecant function which can be converted into other trigonometric functions.

Complete step-by-step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle. The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant sine function and cosecant function are positive and in third only tangent function and cotangent function is positive. The fourth quadrant has only cosine and secant are positive.

The above expression has a cosecant function in it. When the length of the hypotenuse is divided by the length of the opposite side, it gives the cosecant of an angle in a right triangle. It is denoted as csc and sometimes as Cosec.
We now apply the trigonometric identity of cosecant function.
 i.e., \[\cos ecx = \dfrac{1}{{\sin x}}\]
Therefore, we get,
\[\dfrac{1}{{{{\sin }^2}x}} - 1\]
Now on further solving it so that denominators are equal we will get,
\[\dfrac{{1 - {{\sin }^2}x}}{{{{\sin }^2}x}} = \dfrac{{{{\cos }^2}x}}{{{{\sin }^2}x}} = {\cot ^2}x\]
Therefore, we get the above solution.


Note:An important thing to note is that the cotangent function is the opposite of tangent function. The trigonometric identity of tangent function is sine function divided by cosine function so the identity of cotangent function is the opposite i.e., cosine function divided by sine function.