Understanding the Properties and Definition of the Cosecant Function: Evaluating the Expression csc^2(x) and How to Simplify It

int csc^2(x)

The expression “csc^2(x)” represents the square of the cosecant of angle x

The expression “csc^2(x)” represents the square of the cosecant of angle x. To evaluate this expression, we need to understand the properties and definition of the cosecant function.

The cosecant function (csc) is defined as the reciprocal of the sine function. It can be represented as:

csc(x) = 1 / sin(x)

To find csc^2(x), we need to square both sides of this equation. Squaring both sides will eliminate the fraction and give us:

csc^2(x) = (1 / sin(x))^2

To simplify this expression, we square the numerator and denominator separately. The numerator squared is 1^2, which is equal to 1. The denominator squared requires expanding the square:

(sin(x))^2 = sin(x) * sin(x) = sin^2(x)

Therefore, csc^2(x) can be simplified as:

csc^2(x) = (1 / sin(x))^2 = 1 / sin^2(x)

So, the simplified expression for csc^2(x) is 1 / sin^2(x).

It is worth noting that the value of csc^2(x) can also be calculated using the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. Rearranging this equation, we can express sin^2(x) as 1 – cos^2(x). Substituting this into the expression for csc^2(x), we get:

csc^2(x) = 1 / (1 – cos^2(x))

Both forms are equivalent and can be used depending on the context and preference.

More Answers:

Understanding the Integration of sin(x) and Applying Basic Integration Rules
Mastering Integration: A Comprehensive Guide to Finding the Integral of cos(x) with Respect to x
Understanding the Simplified Expression of sec^2(x) in Trigonometry

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