tanA +secA - 1/tanA - secA = secA +tanA = 1+ sinA / cosA

Let's simplify the given expression step by step to verify if:

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

### Step 1: Start with the left-hand side

We will start with the first part of the expression:

\[
\frac{\tan A + \sec A - 1}{\tan A - \sec A}
\]

Recall that:
- \(\tan A = \frac{\sin A}{\cos A}\)
- \(\sec A = \frac{1}{\cos A}\)

Now, substitute \(\tan A\) and \(\sec A\) into the expression:

\[
\frac{\frac{\sin A}{\cos A} + \frac{1}{\cos A} - 1}{\frac{\sin A}{\cos A} - \frac{1}{\cos A}}
\]

Simplify both the numerator and denominator by combining like terms:

Numerator:
\[
\frac{\sin A + 1 - \cos A}{\cos A}
\]

Denominator:
\[
\frac{(\cos

tanA + secA - 1 / tanA - secA = secA + tanA = 1 + sinA / cosA

This step-by-step guide shows how to simplify the trigonometric expression tanA + secA - 1 / tanA - secA and verify its equivalence to secA + tanA = 1 + sinA / cosA using key trigonometric identities. Suitable for high school students in Grades 10-12.

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