Question

Simplify.

cotx⋅secx⋅csc^4x⋅sin^5x



1
sin^4x
tan^2x
sin^4xcos^5x

Answer 1

To simplify the expression \( \cot x \cdot \sec x \cdot \csc^4 x \cdot \sin^5 x \), we will first rewrite the trigonometric functions in terms of sine and cosine.

We know the following identities:

\[
\cot x = \frac{\cos x}{\sin x}, \quad \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}.
\]

Substituting these identities into the expression, we get:

\[
\cot x \cdot \sec x \cdot \csc^4 x \cdot \sin^5 x = \left(\frac{\cos x}{\sin x}\right) \cdot \left(\frac{1}{\cos x}\right) \cdot \left(\frac{1}{\sin^4 x}\right) \cdot \sin^5 x.
\]

Now, simplify it step by step:

1. Multiply \( \cot x \) and \( \sec x \):

\[
\cot x \cdot \sec x = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}.
\]

2. Now substitute this back into the expression:

\[
\frac{1}{\sin x} \cdot \csc^4 x \cdot \sin^5 x = \frac{1}{\sin x} \cdot \frac{1}{\sin^4 x} \cdot \sin^5 x.
\]

3. Simplifying this expression:

\[
\frac{1}{\sin x} \cdot \frac{1}{\sin^4 x} \cdot \sin^5 x = \frac{\sin^5 x}{\sin^5 x} = 1.
\]

Thus, the simplified expression is:

\[
\boxed{1}.
\]