Trigonometry

1. Check periodicity of :

- The term is periodic with a period of .

- However, the term x is not periodic, as it continuously increases without repeating.

- Since the sum of a periodic function ( ) and a non-periodic function (x) cannot be periodic, is not periodic.

2. Check periodicity of :

- The term is not periodic, as it is a continuously increasing function.

- Adding 1 to does not change its non-periodic nature.

- Since depends on a non-periodic term, it is also not periodic.

Thus, the correct answers are (B) and (D).

1. Given:

2. Since and are acute:

• For , and .
• Therefore,

3. Substitute and into the second condition:

4. Verify:

- This holds true.

Thus,

Step 1: The given equation is:

We are also told that:

We need to find the value of .

Step 2: Use the triple angle identity for tangent:

Substitute this into the original equation:

Step 3: Simplify the expression. We now need to express in terms of . Use the double angle identity for tangent:

Substituting this into the equation gives:

Step 4: Now, we are tasked with simplifying and solving this equation. However, notice that we are also given the relationship:

Substituting , we get:

Simplifying:

Step 5: We now solve for . From the structure of the equation, we notice that when , the equation holds. This gives us the value , which is confirmed through further simplification and substitution.

Thus, the value of is: