Trigonometric Identities

We are given the equation:

Step 1: Recall Trigonometric Identity
Using the identity: We'll simplify the right side using this identity.

Step 2: Identifying the Values
From the given equation:

Step 3: Use Identity for Product of Tangents
Using the identity for tangent product, Substituting the known angles,

Step 4: Solving for
By simplifying both sides and using the tangent addition and subtraction identities, the equation simplifies to:

Final Answer: (3)

We are given that in a triangle , Where: - are the exradii opposite to angles respectively.

Step 1: Recall the Exradius Property
In a triangle, Where: - is the area of the triangle - is the semi-perimeter

Step 2: Express the Ratios in Terms of
Since , we assign: Then,

Step 3: Relating with the Sine Rule
By the sine rule in a triangle: This implies: Using the values from Step 2:

Step 4: Simplifying the Ratios
Dividing each term by 1.2:

Final Answer:

We need to analyze two statements:

Step 1: Verifying Statement (S1)
We use the sine addition-subtraction identities: Applying this identity, Now, Since and , Thus, Since , the two sides are close but **not exactly equal**.

Conclusion: (S1) is False. Step 2: Verifying Statement (S2)
Given, Since , - Maximum value of
- Minimum value of
Thus,

Conclusion: (S2) is True. Final Answer: (D) (S1) is false, (S2) is true.