Find the value of `(tanA+secA-1)cosA`.

Options

A. a) `1+sinA`
B. b) `(1+sinA)(tanA-secA+1)`
C. c) `1-sinA`
D. d) None of these

Answer

To find the value of \((\tan A + \sec A - 1) \cos A\), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
- \(\tan A = \frac{\sin A}{\cos A}\)
- \(\sec A = \frac{1}{\cos A}\)

Substituting these into the expression, we have:

\[ (\tan A + \sec A - 1) \cos A = \left(\frac{\sin A}{\cos A} + \frac{1}{\cos A} - 1\right) \cos A \]

### Step 2: Combine the terms inside the parentheses
Now, we can combine the terms inside the parentheses:

\[ = \left(\frac{\sin A + 1 - \cos A}{\cos A}\right) \cos A \]

### Step 3: Multiply by \(\cos A\)
When we multiply by \(\cos A\), the \(\cos A\) in the denominator cancels with the \(\cos A\) outside:

\[ = \sin A + 1 - \cos A \]

### Step 4: Final expression
Thus, the final value of the expression is:

\[ \sin A + 1 - \cos A \]

### Summary
The value of \((\tan A + \sec A - 1) \cos A\) simplifies to \(\sin A + 1 - \cos A\).

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