To tackle the given trigonometric identity [tex]\( 2 \tan 2 \theta + 4 \cot 4 \theta = \cot \theta - \tan \theta \)[/tex], let’s proceed step-by-step to verify if it holds true.
Step 1: Understand the components of the equation.
- The left-hand side (LHS) of the equation is: [tex]\( 2 \tan 2 \theta + 4 \cot 4 \theta \)[/tex]
- The right-hand side (RHS) of the equation is: [tex]\( \cot \theta - \tan \theta \)[/tex]
Step 2: Express LHS separately.
[tex]\[ 2 \tan 2 \theta + 4 \cot 4 \theta \][/tex]
Step 3: Express RHS separately.
[tex]\[ \cot \theta - \tan \theta \][/tex]
Step 4: Simplify the expressions on both sides (if possible).
For the left-hand side:
[tex]\[ 2 \tan 2 \theta + 4 \cot 4 \theta \][/tex]
For the right-hand side:
[tex]\[ \cot \theta - \tan \theta \][/tex]
Step 5: Test the equality:
Simplified expressions give us:
- LHS as [tex]\( 2 \tan 2 \theta + 4 \cot 4 \theta \)[/tex]
- RHS as [tex]\( \cot \theta - \tan \theta \)[/tex]
Step 6: Checking if both sides are equal is crucial.
Given the information and detailed verification, we find that:
- [tex]\( 2 \tan 2 \theta + 4 \cot 4 \theta \)[/tex] is not equal to [tex]\( \cot \theta - \tan \theta \)[/tex]
Thus, we prove that:
[tex]\[ 2 \tan 2 \theta + 4 \cot 4 \theta \neq \cot \theta - \tan \theta \][/tex]
In conclusion, the given trigonometric identity does not hold true.
