Step 1: Mendel's Cross of Pea Plants
Mendel crossed a homozygous long pea plant (dominant, TT) with a homozygous dwarf pea plant (recessive, tt). The F\(_1\) generation consisted of offspring with genotype Tt, where the dominant trait (long plant) expresses itself.
Step 2: Result of the Cross
In Mendel's experiment, all the plants in the F\(_1\) generation were heterozygous (Tt) and exhibited the dominant trait (long plants). The percentage of long plants was 100%. However, in the F\(_2\) generation (produced by self-crossing F\(_1\) plants), the expected ratio of long to dwarf plants was 3:1.
Step 3: Conclusion
Since the F\(_1\) generation consists of 100% long plants (heterozygous Tt), the correct answer is 50% long and 50% dwarf plants in the next generation.
\[
\boxed{\text{50% long and 50% dwarf plants}}
\]
The value of $\dfrac{1+\cot^2 \theta}{1+\tan^2 \theta}$ will be: