Show that $$ \left(1-\sin ^{2} heta ight) \sec ^{2} heta=1 $$ Show that $$ an x+\cot x=\sec x \operatorname{cosec} x $$ Given that $$ \sin x=\frac{5}{13} $$ for $$ 0^{\circ}

Question

Show that $$ \left(1-\sin ^{2} 	hetaight) \sec ^{2} 	heta=1 $$ Show that $$ 	an x+\cot x=\sec x \operatorname{cosec} x $$ Given that $$ \sin x=\frac{5}{13} $$ for $$ 0^{\circ}<	heta<90^{\circ} $$. Find $$ \frac{\cos x-2 \sin x}{2 	an x} $$ Show that $$ \left(\frac{1+\cos 	heta}{1}ight)\left(\frac{\sec 	heta-1}{}ight)=1 $$

UpStudy AI · Accepted Answer

Sure, let's address each of your problems step by step.

---

### **1. Show that $$ \left(1 - \sin^2 \theta\right) \sec^2 \theta = 1 $$**

**Proof:**

1. **Recall the Pythagorean Identity:**
   $$
   \sin^2 \theta + \cos^2 \theta = 1
   $$
   Therefore,
   $$
   1 - \sin^2 \theta = \cos^2 \theta
   $$

2. **Substitute into the Original Expression:**
   $$
   \left(1 - \sin^2 \theta\right) \sec^2 \theta = \cos^2 \theta \cdot \sec^2 \theta
   $$

3. **Simplify Using the Definition of Secant:**
   $$
   \sec \theta = \frac{1}{\cos \theta} \Rightarrow \sec^2 \theta = \frac{1}{\cos^2 \theta}
   $$
   So,
   $$
   \cos^2 \theta \cdot \sec^2 \theta = \cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1
   $$

**Conclusion:**
$$
\left(1 - \sin^2 \theta\right) \sec^2 \theta = 1
$$

---

### **2. Show that $$ \tan x + \cot x = \sec x \csc x $$**

**Proof:**

1. **Express Tangent and Cotangent in Terms of Sine and Cosine:**
   $$
   \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}
   $$

2. **Add the Two Expressions:**
   $$
   \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}
   $$

3. **Find a Common Denominator:**
   $$
   \tan x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}
   $$

4. **Use the Pythagorean Identity:**
   $$
   \sin^2 x + \cos^2 x = 1
   $$
   So,
   $$
   \tan x + \cot x = \frac{1}{\sin x \cos x}
   $$

5. **Express in Terms of Secant and Cosecant:**
   $$
   \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}
   $$
   Therefore,
   $$
   \sec x \csc x = \frac{1}{\cos x} \cdot \frac{1}{\sin x} = \frac{1}{\sin x \cos x}
   $$

**Conclusion:**
$$
\tan x + \cot x = \sec x \csc x
$$

---

### **3. Given that $$ \sin x = \frac{5}{13} $$ for $$ 0^\circ < x < 90^\circ $$. Find $$ \frac{\cos x - 2 \sin x}{2 \tan x} $$**

**Solution:**

1. **Find $$ \cos x $$:**
   Using the Pythagorean identity:
   $$
   \sin^2 x + \cos^2 x = 1 \Rightarrow \left(\frac{5}{13}\right)^2 + \cos^2 x = 1
   $$
   $$
   \frac{25}{169} + \cos^2 x = 1 \Rightarrow \cos^2 x = 1 - \frac{25}{169} = \frac{144}{169}
   $$
   Since $$ 0^\circ < x < 90^\circ $$, $$ \cos x $$ is positive:
   $$
   \cos x = \frac{12}{13}
   $$

2. **Find $$ \tan x $$:**
   $$
   \tan x = \frac{\sin x}{\cos x} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}
   $$

3. **Compute the Numerator ($$ \cos x - 2 \sin x $$):**
   $$
   \cos x - 2 \sin x = \frac{12}{13} - 2 \cdot \frac{5}{13} = \frac{12}{13} - \frac{10}{13} = \frac{2}{13}
   $$

4. **Compute the Denominator ($$ 2 \tan x $$):**
   $$
   2 \tan x = 2 \cdot \frac{5}{12} = \frac{10}{12} = \frac{5}{6}
   $$

5. **Form the Fraction and Simplify:**
   $$
   \frac{\cos x - 2 \sin x}{2 \tan x} = \frac{\frac{2}{13}}{\frac{5}{6}} = \frac{2}{13} \cdot \frac{6}{5} = \frac{12}{65}
   $$

**Conclusion:**
$$
\frac{\cos x - 2 \sin x}{2 \tan x} = \frac{12}{65}
$$

---

### **4. Show that $$ \left(\frac{1 + \cos \theta}{1}\right) \left(\frac{\sec \theta - 1}{\text{?}}\right) = 1 $$**

**Clarification Needed:**

The expression provided seems incomplete because the second fraction $$ \frac{\sec \theta - 1}{\text{?}} $$ is missing a denominator. To provide a meaningful proof, the complete expression is necessary.

However, assuming that the intended expression is:

$$
\left(1 + \cos \theta\right) \left( \sec \theta - 1 \right) = 1
$$

Let's explore this assumption.

**Attempted Proof:**

1. **Express Secant in Terms of Cosine:**
   $$
   \sec \theta = \frac{1}{\cos \theta}
   $$
   So,
   $$
   \sec \theta - 1 = \frac{1}{\cos \theta} - 1 = \frac{1 - \cos \theta}{\cos \theta}
   $$

2. **Multiply with $$ 1 + \cos \theta $$:**
   $$
   (1 + \cos \theta) \left( \sec \theta - 1 \right) = (1 + \cos \theta) \cdot \frac{1 - \cos \theta}{\cos \theta}
   $$
   $$
   = \frac{(1 + \cos \theta)(1 - \cos \theta)}{\cos \theta}
   $$

3. **Use the Difference of Squares:**
   $$
   (1 + \cos \theta)(1 - \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta
   $$
   So,
   $$
   \frac{(1 + \cos \theta)(1 - \cos \theta)}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta}
   $$

4. **Simplify the Expression:**
   $$
   \frac{\sin^2 \theta}{\cos \theta} = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} = \sin \theta \cdot \tan \theta
   $$

5. **Assessing Equality to 1:**
   $$
   \sin \theta \cdot \tan \theta = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta}
   $$
   This expression is **not** generally equal to 1 for all $$ \theta $$. It only equals 1 for specific angles where $$ \frac{\sin^2 \theta}{\cos \theta} = 1 $$, which does not hold true universally.

**Conclusion:**

Given the current form of the expression, $$ \left(1 + \cos \theta\right) \left( \sec \theta - 1 \right) $$ does **not** simplify to 1 for all $$ \theta $$. If there's a different intended expression or additional context, please provide the complete details so that a proper proof can be constructed.

---

If you have any further questions or need clarification on the fourth problem, feel free to ask!
To solve the given problems: 1. **Prove that $$ \left(1 - \sin^2 \theta\right) \sec^2 \theta = 1 $$:** $$ \left(1 - \sin^2 \theta\right) \sec^2 \theta = 1 $$ 2. **Prove that $$ \tan x + \cot x = \sec x \csc x $$:** $$ \tan x + \cot x = \sec x \csc x $$ 3. **Given $$ \sin x = \frac{5}{13} $$ for $$ 0^\circ < x < 90^\circ $$, find $$ \frac{\cos x - 2 \sin x}{2 \tan x} $$:** $$ \frac{\cos x - 2 \sin x}{2 \tan x} = \frac{12}{65} $$ 4. **Prove that $$ \left(1 + \cos \theta\right) \left( \sec \theta - 1 \right) = 1 $$:** The expression is incomplete. Please provide the full expression for a complete proof. **Summary:** - The first two proofs are straightforward trigonometric identities. - The third problem involves calculating a specific trigonometric expression given the value of $$ \sin x $$. - The fourth problem requires the complete expression to be proven. If you need further assistance or have additional questions, feel free to ask!

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