2) Which one of the following expressions is not identical to any of the others?
A $\frac{1}{1+\tan ^{2} \theta}$
B $1+\frac{1}{\operatorname{cosec}^{2} \theta}$
C $\sin ^{2} \theta\left(\operatorname{cosec}^{2} \theta-1\right)$
D $\frac{1}{2}(1+\cos 2 \theta)$
[1 mark]

Step 1 ：Given expressions:

Step 2 ：A $\frac{1}{1+\tan ^{2} \theta}$

Step 3 ：B $1+\frac{1}{\operatorname{cosec}^{2} \theta}$

Step 4 ：C $\sin ^{2} \theta\left(\operatorname{cosec}^{2} \theta-1\right)$

Step 5 ：D $\frac{1}{2}(1+\cos 2 \theta)$

Step 6 ：Simplify each expression:

Step 7 ：A $\cos^2 \theta$

Step 8 ：B $2 + \tan^{-2} \theta$

Step 9 ：C $\cos^2 \theta$

Step 10 ：D $\cos^2 \theta$

Step 11 ：\(\boxed{\text{Final Answer: The expression that is not identical to any of the others is B.}}\)