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Q.24 If tan A = sqrt(2) - 1 then find the value of sinAcosA + tanA.

We are given that \( \tan A = \sqrt{2} - 1 \), and we need to find the value of \( \sin A \cos A + \tan A \).

### Step 1: Express \( \tan A \)
We know:
\[
\tan A = \frac{\sin A}{\cos A}
\]
So, using this, we can say:
\[
\frac{\sin A}{\cos A} = \sqrt{2} - 1
\]
This means:
\[
\sin A = (\sqrt{2} - 1) \cos A
\]

### Step 2: Use the identity \( \sin^2 A + \cos^2 A = 1 \)
Now substitute \( \sin A = (\sqrt{2} - 1) \cos A \) into the identity:
\[
\sin^2 A + \cos^2 A = 1
\]
\[
((\sqrt{2} - 1) \cos A)^2 + \cos^2 A = 1
\]
Simplify:
\[
(\sqrt{2} - 1)^2 \cos^2 A + \cos^2 A = 1
\]
\[
((2 - 2\sqrt{2} + 1) + 1) \cos^2 A = 1
\]
\[
(3 - 2\sqrt{2}) \cos^2 A = 1
\]
\[
\cos^2 A = \frac{1}{3 - 2\sqrt{2}}
\]

### Step 3: Use the value of \( \cos^2 A \) to find \( \sin A \cos A + \tan A \)
From \( \sin A = (\sqrt{2} - 1) \cos A \), the expression \( \sin A \cos A \) becomes:
\[
\sin A \cos A = (\sqrt{2} - 1) \cos^2 A
\]

So, the value we need to compute is:
\[
\sin A \cos A + \tan A = (\sqrt{2} - 1) \cos^2 A + \tan A
\]
Now we know \( \tan A = \sqrt{2} - 1 \), so we can write this as:
\[
(\sqrt{2} - 1) \cos^2 A + (\sqrt{2} - 1)
\]

### Step 4: Factor the expression
\[
(\sqrt{2} - 1)(\cos^2 A + 1)
\]
Since \( \cos^2 A + \sin^2 A = 1 \), we know that \( \cos^2 A + 1 = 1 + 1 = 2 \). So the expression simplifies to:
\[
(\sqrt{2} - 1) \times 2 = 2\sqrt{2} - 2
\]

Thus, the value of \( \sin A \cos A + \tan A \) is:
\[
\boxed{2\sqrt{2} - 2}
\]

Would you like further explanation on any part of this process?

Here are 5 related questions:
1. How do you solve trigonometric equations involving both \( \sin A \) and \( \cos A \)?
2. What is the Pythagorean identity and how is it used in trigonometry?
3. How do you convert between trigonometric functions like \( \tan A \), \( \sin A \), and \( \cos A \)?
4. What are some real-life applications of trigonometric identities?
5. How can you simplify complex trigonometric expressions efficiently?

**Tip:** Always check if an expression can be factored, as this can significantly simplify your calculations.

Q.24 If tan A = sqrt(2) - 1 then find the value of sinAcosA + tanA.

Solve the trigonometric problem where tan A = sqrt(2) - 1, and find the value of sinAcosA + tanA. This problem covers key trigonometric concepts like the Pythagorean identity and algebraic manipulation. Suitable for students in Grades 9-11.

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