To solve the equation [tex]\(\frac{1}{\csc^2 x} + \cos^2 x = 1\)[/tex], let's break it down step by step.
1. Understand the Terms:
- [tex]\(\csc x\)[/tex] is the cosecant of [tex]\(x\)[/tex], which is the reciprocal of the sine function: [tex]\(\csc x = \frac{1}{\sin x}\)[/tex].
- Therefore, [tex]\(\csc^2 x = \left( \frac{1}{\sin x} \right)^2 = \frac{1}{\sin^2 x}\)[/tex].
2. Rewrite the Equation:
Substitute [tex]\(\csc^2 x\)[/tex] in the equation:
[tex]\[
\frac{1}{\csc^2 x} = \sin^2 x
\][/tex]
So, the original equation can be written as:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
3. Use the Pythagorean Identity:
Recall the fundamental Pythagorean identity in trigonometry:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
4. Compare the Equations:
Notice that our restructured equation [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex] is exactly the same as the established Pythagorean identity. Hence, the equation is always true for all [tex]\(x\)[/tex].
5. Conclusion:
Therefore, simplifying the given trigonometric equation [tex]\(\frac{1}{\csc^2 x} + \cos^2 x = 1\)[/tex] results in the identity [tex]\(1 = 1\)[/tex], which simplifies to:
[tex]\[
0 = 0
\][/tex]
This confirms that the given equation holds true for all values of [tex]\(x\)[/tex]. Hence, the simplified form of the given equation is [tex]\(0\)[/tex].
