Which of the following is/are +ve ?

Options

A. `log_(sin1) tan1`
B. `log_(cos1)(1+tan3)`
C. `log_(log_(10)5)(cos theta+sec theta)`
D. `log_(tan15^(@))(2 sin18^(@))`

Answer

To determine which of the given options are positive, we will analyze each option using properties of logarithms and the values of trigonometric functions.

### Step-by-Step Solution:

1. **Option 1: log(tan(1)) to the base sin(1)**
- We need to check the values of tan(1) and sin(1).
- **tan(1 radian)** is greater than 1.
- **sin(1 radian)** is less than 1.
- Since tan(1) > 1 and sin(1) < 1, they lie on opposite sides of 1.
- Therefore, log(tan(1)) to the base sin(1) is negative.
- **Conclusion:** This option is rejected.

2. **Option 2: log(1 + tan(3)) to the base cos(1)**
- We need to evaluate 1 + tan(3).
- **tan(3 radians)** is negative, hence 1 + tan(3) < 1.
- **cos(1 radian)** is also less than 1.
- Since both values (1 + tan(3) and cos(1)) are less than 1, they lie on the same side of 1.
- Therefore, log(1 + tan(3)) to the base cos(1) is positive.
- **Conclusion:** This option is accepted.

3. **Option 3: log(cos(θ) + sec(θ)) to the base log(5) to the base 10**
- We can express cos(θ) + sec(θ) as (cos²(θ) + 1) / cos(θ).
- The numerator (cos²(θ) + 1) is always greater than 1 (since cos²(θ) is between 0 and 1).
- The denominator (cos(θ)) is also between 0 and 1.
- Thus, cos(θ) + sec(θ) > 1.
- **log(5) to the base 10** is less than 1.
- Since we have one value greater than 1 and the other less than 1, this logarithm will be negative.
- **Conclusion:** This option is rejected.

4. **Option 4: log(sin(18)) to the base tan(15)**
- We know that sin(18) is approximately (√5 - 1) / 4, which is slightly less than 1.
- tan(15) is also less than 1.
- Since both sin(18) and tan(15) are less than 1, they lie on the same side of 1.
- Therefore, log(sin(18)) to the base tan(15) is positive.
- **Conclusion:** This option is accepted.

### Final Answer:
The options that are positive are **Option 2** and **Option 4**.