verify the identity sect+1/tant=tant/sect-1 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 given trigonometric identity:

\[
\frac{\sec t + 1}{\tan t} = \frac{\tan t}{\sec t - 1}
\]

### Step 1: Begin with the left-hand side (LHS)

\[
\text{LHS} = \frac{\sec t + 1}{\tan t}
\]

Recall that \(\sec t = \frac{1}{\cos t}\) and \(\tan t = \frac{\sin t}{\cos t}\).

So, we substitute these into the LHS:

\[
\frac{\sec t + 1}{\tan t} = \frac{\frac{1}{\cos t} + 1}{\frac{\sin t}{\cos t}}
\]

### Step 2: Simplify the LHS

The expression in the numerator simplifies as follows:

\[
\frac{1}{\cos t} + 1 = \frac{1 + \cos t}{\cos t}
\]

Now the LHS becomes:

\[
\frac{\frac{1 + \cos t}{\cos t}}{\frac{\sin t}{\cos t}} = \frac{1 + \cos t}{\sin t}
\]

### Step 3: Work on the right-hand side (RHS)

Now let's examine the right-hand side (RHS):

\[
\text{RHS} = \frac{\tan t}{\sec t - 1}
\]

Substitute the values of \(\sec t = \frac{1}{\cos t}\) and \(\tan t = \frac{\sin t}{\cos t}\):

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

Simplify the denominator:

\[
\frac{1}{\cos t} - 1 = \frac{1 - \cos t}{\cos t}
\]

So the RHS becomes:

\[
\frac{\frac{\sin t}{\cos t}}{\frac{1 - \cos t}{\cos t}} = \frac{\sin t}{1 - \cos t}
\]

### Step 4: Compare LHS and RHS

At this point, we have:

\[
\text{LHS} = \frac{1 + \cos t}{\sin t}
\]
and
\[
\text{RHS} = \frac{\sin t}{1 - \cos t}
\]

These two expressions are not equal, so the given equation is **not** an identity.

Would you like to explore why these expressions differ, or how such problems could be adjusted? Here are 5 related questions to deepen your understanding:

1. How do reciprocal identities like \(\sec t\) and \(\csc t\) simplify trigonometric equations?
2. What is the importance of factoring when solving trigonometric identities?
3. Can we apply Pythagorean identities here to further simplify expressions?
4. How do you determine if a trigonometric equation is a conditional equation rather than an identity?
5. What are common mistakes to avoid when verifying trigonometric identities?

**Tip:** Always check for possible simplifications using fundamental identities (Pythagorean, reciprocal, and quotient) before moving forward in verifying complex identities.

verify the identity sect+1/tant=tant/sect-1 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 of the trigonometric identity sec(t) + 1 / tan(t) = tan(t) / (sec(t) - 1). Learn the application of reciprocal identities and Pythagorean identities, simplifying trigonometric equations. Suitable for Grades 10-12.

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