Consider the following sttements:
1. `cos theta + sec theta` can never be equal to 1.5
2. `tan theta + cot theta` can never be less than 2

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the given statements, we will analyze each one step by step.

### Step 1: Analyze the first statement: `cos θ + sec θ = 1.5`

1. We can rewrite the expression using the definition of secant:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Therefore, the expression becomes:
\[ \cos \theta + \frac{1}{\cos \theta} \]

2. Let \( x = \cos \theta \). The expression now is:
\[ x + \frac{1}{x} \]

3. We know from the AM-GM inequality that:
\[ x + \frac{1}{x} \geq 2 \]
for \( x > 0 \). The minimum value occurs when \( x = 1 \) (i.e., \( \theta = 0 \)).

4. Since \( x + \frac{1}{x} \) is always greater than or equal to 2, it can never be equal to 1.5.

### Conclusion for the first statement:
The first statement is **true**.

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### Step 2: Analyze the second statement: `tan θ + cot θ < 2`

1. We can rewrite the expression using the definition of cotangent:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Therefore, the expression becomes:
\[ \tan \theta + \frac{1}{\tan \theta} \]

2. Let \( y = \tan \theta \). The expression now is:
\[ y + \frac{1}{y} \]

3. Again, using the AM-GM inequality:
\[ y + \frac{1}{y} \geq 2 \]
for \( y > 0 \). The minimum value occurs when \( y = 1 \) (i.e., \( \theta = 45^\circ \)).

4. Since \( y + \frac{1}{y} \) is always greater than or equal to 2, it can never be less than 2.

### Conclusion for the second statement:
The second statement is **true**.

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### Final Conclusion:
Both statements are true.

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