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Solution
1) sin x + cos x + sin x cos x = 1
1
(sin x + cos x)^2 = sin^2 x + cos^2 x + 2sin x cos x
(sin x + cos x)^2 = 1 + 2sin x cos x
∴ sin xcos x = ((sin x + cos x)^2 - 1) / 2
1) => sin x + cos x + ((sin x + cos x)^2 - 1) / 2 = 1
=> (sin x + cos x)^2 + 2(sin x + cos x) - 3 = 0
We let u = sin x + cos x , -1 ≤ u ≤ 1
=> u^2 + 2u - 3 = 0
Hence, (u - 1)(u + 3) = 0
=> u = 1 or u = -3
∴ u = 1 is valid in the required
interval above.
so, sin x + cos x = 1
√2 sin (x + π/4) = 1
=> sin (x + π/4) = 1/√2
∴ p.v = sin^-1 (1/√2) = π/4
General soln : x + π/4 = πn + (-1)^n p.v
Hence, x = -π/4 + (-1)^n π/4 + πn, n ∈ Z
Mathematics
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√x^2 = |x|
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Results
1. **State the problem:** Solve the equation $$\sin x + \cos x + \sin x \cos x = 1$$.
2. **Use the identity:** Recall that $$(\sin x + \cos x)^2 = \sin^2 x + \cos^2 x + 2 \sin x \cos x = 1 + 2 \sin x \cos x$$.
3. **Express $\sin x \cos x$ in terms of $\sin x + \cos x$:**
$$\sin x \cos x = \frac{(\sin x + \cos x)^2 - 1}{2}$$
4. **Substitute back into the original equation:**
$$\sin x + \cos x + \frac{(\sin x + \cos x)^2 - 1}{2} = 1$$
5. **Multiply both sides by 2 to clear the denominator:**
$$2(\sin x + \cos x) + (\sin x + \cos x)^2 - 1 = 2$$
6. **Rearrange:**
$$(\sin x + \cos x)^2 + 2(\sin x + \cos x) - 3 = 0$$
7. **Let $u = \sin x + \cos x$ with $-1 \leq u \leq 1$:**
$$u^2 + 2u - 3 = 0$$
8. **Factor the quadratic:**
$$(u - 1)(u + 3) = 0$$
9. **Solve for $u$:**
$$u = 1 \quad \text{or} \quad u = -3$$
10. **Check domain:** Since $u = \sin x + \cos x$ and $-1 \leq u \leq 1$, only $u = 1$ is valid.
11. **Solve $\sin x + \cos x = 1$:**
Rewrite as:
$$\sqrt{2} \sin \left(x + \frac{\pi}{4}\right) = 1$$
12. **Divide both sides by $\sqrt{2}$:**
$$\sin \left(x + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$$
13. **Find principal value:**
$$\text{p.v.} = \sin^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}$$
14. **Write general solution:**
$$x + \frac{\pi}{4} = \pi n + (-1)^n \frac{\pi}{4}, \quad n \in \mathbb{Z}$$
15. **Isolate $x$:**
$$x = -\frac{\pi}{4} + (-1)^n \frac{\pi}{4} + \pi n, \quad n \in \mathbb{Z}$$
**Final answer:**
$$x = -\frac{\pi}{4} + (-1)^n \frac{\pi}{4} + \pi n, \quad n \in \mathbb{Z}$$