Let's solve the given equation step by step:
[tex]\[
\sqrt{1 - \sin^2(A)} \cdot \csc(A) = \cot(A)
\][/tex]
First, we need to use some fundamental trigonometric identities to simplify this expression.
1. Pythagorean Identity: We know that for any angle [tex]\(A\)[/tex],
[tex]\[
\sin^2(A) + \cos^2(A) = 1
\][/tex]
Therefore,
[tex]\[
\cos^2(A) = 1 - \sin^2(A)
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
\sqrt{1 - \sin^2(A)} = \cos(A)
\][/tex]
2. Reciprocal Identities:
- The cosecant function is the reciprocal of the sine function:
[tex]\[
\csc(A) = \frac{1}{\sin(A)}
\][/tex]
- The cotangent function is defined as the ratio of cosine to sine:
[tex]\[
\cot(A) = \frac{\cos(A)}{\sin(A)}
\][/tex]
Now, substitute these identities back into the given equation:
The left side of the equation:
[tex]\[
\sqrt{1 - \sin^2(A)} \cdot \csc(A)
\][/tex]
becomes:
[tex]\[
\cos(A) \cdot \frac{1}{\sin(A)}
\][/tex]
which simplifies to:
[tex]\[
\frac{\cos(A)}{\sin(A)}
\][/tex]
So, the equation now is:
[tex]\[
\frac{\cos(A)}{\sin(A)} = \cot(A)
\][/tex]
Since [tex]\(\frac{\cos(A)}{\sin(A)}\)[/tex] is indeed the definition of [tex]\(\cot(A)\)[/tex], the given equation holds true. Thus, we have shown that:
[tex]\[
\sqrt{1 - \sin^2(A)} \cdot \csc(A) = \cot(A)
\][/tex]
This confirms that the given equation is a true trigonometric identity.
