Sure, let us verify the given equation step-by-step:
[tex]\[ \frac{(1+\cos \beta)+(1-\cos \beta)^2}{\sin^2 \beta} = 2\left(1+2 \cot^2 \beta\right) \][/tex]
### Step-by-Step Solution:
1. Simplify the Left-Hand Side (LHS):
[tex]\[
\frac{(1+\cos \beta)+(1-\cos \beta)^2}{\sin^2 \beta}
\][/tex]
Start with simplifying the numerator:
[tex]\[
(1+\cos \beta) + (1 - \cos \beta)^2
\][/tex]
Expand [tex]\((1 - \cos \beta)^2\)[/tex]:
[tex]\[
(1 - \cos \beta)^2 = 1 - 2\cos \beta + \cos^2 \beta
\][/tex]
Therefore, the numerator becomes:
[tex]\[
(1 + \cos \beta) + 1 - 2\cos \beta + \cos^2 \beta
\][/tex]
Combine like terms:
[tex]\[
1 + \cos \beta + 1 - 2\cos \beta + \cos^2 \beta = \cos^2 \beta - \cos \beta + 2
\][/tex]
Now, let's write the entire fraction:
[tex]\[
\frac{\cos^2 \beta - \cos \beta + 2}{\sin^2 \beta}
\][/tex]
2. Simplify the Right-Hand Side (RHS):
[tex]\[
2 \left(1 + 2 \cot^2 \beta\right)
\][/tex]
Recall that:
[tex]\[
\cot \beta = \frac{\cos \beta}{\sin \beta}
\][/tex]
Therefore:
[tex]\[
\cot^2 \beta = \left(\frac{\cos \beta}{\sin \beta}\right)^2 = \frac{\cos^2 \beta}{\sin^2 \beta}
\][/tex]
Substitute [tex]\(\cot^2 \beta\)[/tex] back into the equation:
[tex]\[
2 \left(1 + 2 \frac{\cos^2 \beta}{\sin^2 \beta}\right)
\][/tex]
Distribute the 2:
[tex]\[
2 + 4 \frac{\cos^2 \beta}{\sin^2 \beta} = 2 + 4 \cot^2 \beta
\][/tex]
3. Compare LHS and RHS:
LHS:
[tex]\[
\frac{\cos^2 \beta - \cos \beta + 2}{\sin^2 \beta}
\][/tex]
RHS:
[tex]\[
2 + 4 \cot^2 \beta = 2 + 4 \frac{\cos^2 \beta}{\sin^2 \beta}
\][/tex]
Rewrite RHS with the common denominator [tex]\(\sin^2 \beta \)[/tex]:
[tex]\[
2 + 4 \frac{\cos^2 \beta}{\sin^2 \beta} = \frac{2 \sin^2 \beta + 4 \cos^2 \beta}{\sin^2 \beta}
\][/tex]
Now, the comparison is between:
[tex]\[
\frac{\cos^2 \beta - \cos \beta + 2}{\sin^2 \beta} \quad \text{and} \quad \frac{2 \sin^2 \beta + 4 \cos^2 \beta}{\sin^2 \beta}
\][/tex]
4. Conclusion:
Upon simplification, the fractions on both sides are not equal, and therefore:
[tex]\[
\frac{(\cos^2 \beta - \cos \beta + 2)}{\sin^2 \beta} \neq 2 + 4 \cot^2 \beta
\][/tex]
Hence, the given equation is not an identity. The left-hand side does not equate to the right-hand side for all values of [tex]\(\beta\)[/tex].
