Solve the following equations for `0^(@)lethetale90^(@)` :
(i) `2cos^(2)theta=(1)/(2)`
(ii) `4sin^(2)theta-3=0`
(iii) `sin^(2)theta-(1)/(2)sintheta=0`
(iv) `tan^(2)theta-(sqrt(3)+1)tantheta+sqrt(3)=0`
(v) `sintheta-costheta=0`

Answer

Let's solve the given equations step by step for \(0^\circ \leq \theta \leq 90^\circ\).

### (i) \(2\cos^2\theta = \frac{1}{2}\)

1. **Divide both sides by 2:**
\[ \cos^2\theta = \frac{1}{4} \]

2. **Take the square root of both sides:**
\[ \cos\theta = \pm \frac{1}{2} \]

3. **Find the angles:**
- Since \(\cos\theta = \frac{1}{2}\), \(\theta = 60^\circ\).
- We discard \(\cos\theta = -\frac{1}{2}\) because it is not in the range \(0^\circ \leq \theta \leq 90^\circ\).

**Solution:** \(\theta = 60^\circ\)

### (ii) \(4\sin^2\theta - 3 = 0\)

1. **Rearrange the equation:**
\[ 4\sin^2\theta = 3 \]

2. **Divide both sides by 4:**
\[ \sin^2\theta = \frac{3}{4} \]

3. **Take the square root of both sides:**
\[ \sin\theta = \pm \frac{\sqrt{3}}{2} \]

4. **Find the angles:**
- Since \(\sin\theta = \frac{\sqrt{3}}{2}\), \(\theta = 60^\circ\).
- We discard \(\sin\theta = -\frac{\sqrt{3}}{2}\) because it is not in the range \(0^\circ \leq \theta \leq 90^\circ\).

**Solution:** \(\theta = 60^\circ\)

### (iii) \(\sin^2\theta - \frac{1}{2}\sin\theta = 0\)

1. **Factor the equation:**
\[ \sin\theta(\sin\theta - \frac{1}{2}) = 0 \]

2. **Set each factor to zero:**
- \(\sin\theta = 0\) gives \(\theta = 0^\circ\).
- \(\sin\theta - \frac{1}{2} = 0\) gives \(\sin\theta = \frac{1}{2}\), which leads to \(\theta = 30^\circ\).

**Solution:** \(\theta = 0^\circ, 30^\circ\)

### (iv) \(\tan^2\theta - (\sqrt{3}+1)\tan\theta + \sqrt{3} = 0\)

1. **Let \(x = \tan\theta\):**
\[ x^2 - (\sqrt{3}+1)x + \sqrt{3} = 0 \]

2. **Use the quadratic formula:**
\[ x = \frac{-(\sqrt{3}+1) \pm \sqrt{(\sqrt{3}+1)^2 - 4\cdot1\cdot\sqrt{3}}}{2\cdot1} \]

3. **Calculate the discriminant:**
\[ (\sqrt{3}+1)^2 - 4\sqrt{3} = 3 + 2\sqrt{3} + 1 - 4\sqrt{3} = 4 - 2\sqrt{3} \]

4. **Substituting back:**
\[ x = \frac{\sqrt{3}+1 \pm \sqrt{4 - 2\sqrt{3}}}{2} \]

5. **Calculate the values of \(x\) (i.e., \(\tan\theta\)) and find \(\theta\):**
- Solve for \(x\) and then find \(\theta\) using \(\tan^{-1}(x)\).

**Solution:** \(\theta = 60^\circ, 45^\circ\)

### (v) \(\sin\theta - \cos\theta = 0\)

1. **Rearrange the equation:**
\[ \sin\theta = \cos\theta \]

2. **Find the angle where sine and cosine are equal:**
- This occurs at \(\theta = 45^\circ\).

**Solution:** \(\theta = 45^\circ\)

### Summary of Solutions:
1. \( \theta = 60^\circ \)
2. \( \theta = 60^\circ \)
3. \( \theta = 0^\circ, 30^\circ \)
4. \( \theta = 60^\circ, 45^\circ \)
5. \( \theta = 45^\circ \)