Sure! Let's break down and verify the equation:

$\frac{1}{\cot x} + \tan x = \frac{\sin x}{\sec x}$

Step 1: Express $\cot x$ and $\tan x$ in terms of sine and cosine

We know the following trigonometric identities: $\cot x = \frac{\cos x}{\sin x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x}$

So, we can rewrite the left-hand side of the equation:

$\frac{1}{\cot x} = \frac{1}{\frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x}$

Thus, the left-hand side becomes:

$\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x} = 2 \cdot \frac{\sin x}{\cos x}$

Step 2: Simplify the right-hand side

Now let's simplify the right-hand side of the equation. Recall the identity $\sec x = \frac{1}{\cos x}$, so:

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

Step 3: Compare both sides

We now have:

• Left-hand side: $2 \cdot \frac{\sin x}{\cos x}$
• Right-hand side: $\sin x \cdot \cos x$

The equation becomes: $2 \cdot \frac{\sin x}{\cos x} = \sin x \cdot \cos x$

Step 4: Check if both sides are equal

Let’s try to simplify and see if both sides are equal:

• Left-hand side: $2 \cdot \frac{\sin x}{\cos x} = \frac{2 \sin x}{\cos x}$
• Right-hand side: $\sin x \cdot \cos x$

Clearly, the two sides are not equivalent for all values of $x$. The equation doesn't hold true for all values of $x$.

Conclusion

The given equation:

$\frac{1}{\cot x} + \tan x = \frac{\sin x}{\sec x}$

is not valid for all values of $x$.

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If you would like more details or have any questions, feel free to ask!

Here are some related questions:

1. Can you explain why $\frac{1}{\cot x}$ is equivalent to $\tan x$?
2. How do you prove trigonometric identities using a different method, like common denominators?
3. What other common trigonometric identities should be memorized for simplification?
4. How would the equation change if we had $\sec x + \tan x$ on the left-hand side instead of $\frac{1}{\cot x} + \tan x$?
5. What are the specific values of $x$ for which the given equation could hold true?

Tip: When verifying or proving trigonometric identities, always try to express all terms in terms of sine and cosine, as this often simplifies the problem.