verify the identity sinx-tanx/cosx-1=tanx to verify that an equation is an identity, star with more complicated side and transform the side of the equation into the other side by a sequence of steps, each of which produces an identity. begin with left side

Let's verify the identity step by step, starting with the left-hand side (LHS) of the equation:

\[
\sin x - \frac{\tan x}{\cos x - 1} = \tan x
\]

### Step 1: Rewrite \(\tan x\) in terms of sine and cosine
We know that:
\[
\tan x = \frac{\sin x}{\cos x}
\]
So the equation becomes:

\[
\sin x - \frac{\frac{\sin x}{\cos x}}{\cos x - 1}
\]

### Step 2: Simplify the second term
We have a complex fraction. Simplify by multiplying the numerator and denominator by \(\cos x\):

\[
\sin x - \frac{\sin x}{\cos x (\cos x - 1)}
\]

### Step 3: Combine terms into a single fraction
To subtract the two terms, we need a common denominator. Rewrite \(\sin x\) as \(\frac{\sin x(\cos x (\cos x - 1))}{\cos x (\cos x - 1)}\):

\[
\frac{\sin x (\cos x (\cos x - 1))}{\cos x (\cos x - 1)} - \frac{\sin x}{\cos x (\cos x - 1)}
\]

Now combine them into a single fraction:

\[
\frac{\sin x \cos^2 x - \sin x}{\cos x (\cos x - 1)}
\]

### Step 4: Factor out \(\sin x\) from the numerator
Factor out \(\sin x\) from the numerator:

\[
\frac{\sin x (\cos^2 x - 1)}{\cos x (\cos x - 1)}
\]

### Step 5: Use the Pythagorean identity \(\cos^2 x - 1 = -\sin^2 x\)
Using the identity \(\cos^2 x - 1 = -\sin^2 x\), the expression becomes:

\[
\frac{\sin x (-\sin^2 x)}{\cos x (\cos x - 1)} = \frac{-\sin^3 x}{\cos x (\cos x - 1)}
\]

### Step 6: Compare with the right-hand side (RHS)
Recall that the right-hand side of the original equation is \(\tan x = \frac{\sin x}{\cos x}\).

Thus, we need to factor out \(\sin x\) from both the numerator and denominator to match the RHS. However, after simplification, this equation does not simplify to match the right-hand side.

### Conclusion:
After simplifying the left-hand side, the expressions do not appear to be identical. Therefore, the equation provided:

\[
\sin x - \frac{\tan x}{\cos x - 1} = \tan x
\]

is **not** an identity.

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Do you want further details or have any questions?

Here are 5 related questions to expand this topic:
1. What are some common trigonometric identities?
2. How do we simplify complex trigonometric expressions?
3. What is the process of verifying trigonometric identities in general?
4. Can we prove other identities using similar techniques, like \( \sin^2x + \cos^2x = 1 \)?
5. How do the Pythagorean identities help simplify trigonometric expressions?

**Tip:** Always apply known trigonometric identities like the Pythagorean identities when simplifying expressions.

Verify the identity sinx - tanx/cosx - 1 = tanx. To verify that an equation is an identity, start with the more complicated side and transform the side of the equation into the other side by a sequence of steps, each of which produces an identity. Begin with the left side.

Explore the step-by-step verification process of the trigonometric identity sin(x) - tan(x)/(cos(x) - 1) = tan(x). This detailed solution covers key trigonometric concepts, such as rewriting trigonometric functions, using the Pythagorean identity, and simplifying complex fractions. Suitable for high school students in Grades 10-12.

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