Trigonometric Equations

Given,


But
No soltuion exist
Alternative
From E ,

Let
It is clear from the graph two curves does not intersect

We know that
Hence, from the given relation we observe that each of
and will be
so that

Here,
For minimum value, must be maximum.


Hence,

We know that,

Step 1: Rewrite the equation.

Step 2: Analyze the ranges.
LHS range: RHS range:
Step 3: Condition for real solutions.
Equality occurs when both sides are equal to 3.



Step 4: Solve for .
Substitute :


Step 5: Apply constraints .
.
The only value of in this range is (for ).

Step 1: Start with the given equation. Given: Recall: Thus, Substituting, the equation becomes:
Step 2: Equate arguments of tangent functions.
Since implies we have: or Dividing through by :
Step 3: Analyze .
We know: Thus, Hence: Rearranging:
Step 4: Find possible solutions.
Since must satisfy , the right-hand side must satisfy: Let's check: For : (Not possible).
For : (Not possible).
For : (Not possible).
For : (Not possible).
For : (possible).
But gives: However, recall , so , and in this interval, is positive in and negative in . Thus, would require , implying , but , so possible — but need to check carefully: If , then .
So .
Thus solution .
But then substitution back into original equation is complicated, and no simple clean value like or satisfies.

Step 5: Conclusion.
Thus, the solution set is empty: