Question

True or false? tanx⋅cscx⋅cosx⋅cotx
=
cotx
is an identity.



False

True

Answer 1

To determine whether the equation \( \tan x \cdot \csc x \cdot \cos x \cdot \cot x = \cot x \) is an identity, we can simplify the left side.

Recall the definitions of the trigonometric functions involved:

- \( \tan x = \frac{\sin x}{\cos x} \)
- \( \csc x = \frac{1}{\sin x} \)
- \( \cos x = \cos x \)
- \( \cot x = \frac{\cos x}{\sin x} \)

Now, substituting these definitions into the left side:

\[
\tan x \cdot \csc x \cdot \cos x \cdot \cot x = \left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{1}{\sin x}\right) \cdot \cos x \cdot \left(\frac{\cos x}{\sin x}\right)
\]

We can simplify this step by step:

1. \( \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} = \frac{1}{\cos x} \)
2. Then, multiplying this result by \( \cos x \):
\[
\frac{1}{\cos x} \cdot \cos x = 1
\]
3. Now, multiply this by \( \frac{\cos x}{\sin x} \):
\[
1 \cdot \frac{\cos x}{\sin x} = \frac{\cos x}{\sin x} = \cot x
\]

Thus, the left side simplifies to \( \cot x \).

Therefore, we have:

\[
\tan x \cdot \csc x \cdot \cos x \cdot \cot x = \cot x
\]

This shows that the original expression is indeed an identity.

The answer is **True**.