Question

prove the identity.

\\\cot x \left(1 + \tan^2 x\
ight) = \frac{1}{\sin x \cos x}\\

click here to learn more about writing a trigonometric proof.

statement | reason
--- | ---
\\(\cot x \left(1 + \tan^2 x\
ight)\\) |
\\(=\\) | select

Explanation:

Apply Pythagorean identity

Using the Pythagorean Identities knowledge point

$$ 1 + \tan^2 x = \sec^2 x $$

$$ \cot x (1 + \tan^2 x) = \cot x \sec^2 x $$

Convert to sine and cosine

Using the Quotient Identities and Reciprocal Identities knowledge points

$$ \cot x = \frac{\cos x}{\sin x}, \quad \sec^2 x = \frac{1}{\cos^2 x} $$

$$ \cot x \sec^2 x = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^2 x} $$

Simplify the expression

Using the Trigonometric Identities knowledge point

$$ \frac{\cos x}{\sin x \cdot \cos^2 x} = \frac{1}{\sin x \cos x} $$

Answer:

To prove the identity \(\cot x (1 + \tan^2 x) = \frac{1}{\sin x \cos x}\), the first step in the proof table is:

• Statement: \(\cot x \sec^2 x\)
• Reason: Pythagorean Identity