If cot 1 sin 4m and cot 1 minus sin 4n then which one of the following is correct ........

Quantitative Aptitude (CDS) Trignometry

If cot $\Theta$ (1 + sin $\Theta$ ) = 4m and cot $\Theta$ (1 − sin $\Theta$ ) = 4n, then which one of the following is correct?

(m² + n²)² = mn

(m² - n²)² = mn

(m² - n²)² = m²n²

(m² + n²)² = m²n²

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Explanation:

We can write that:
m=cot $\Theta$ (1+sin $\Theta$ )/4
n=cot $\Theta$ (1-sin $\Theta$ )/4
Then, mn=[cot $\Theta$ (1+sin $\Theta$ )/4][cot $\Theta$ (1-sin $\Theta$ )/4]
mn=cot ² $\Theta$ (1-sin ² $\Theta$ )/16
(using cot ² $\Theta$ =cos ² $\Theta$ /sin ² $\Theta$ and 1-sin ² $\Theta$ =cos ² $\Theta$ )
mn=[(cos ² $\Theta$ /sin ² $\Theta$ )* cos ² $\Theta$ ]/16
mn= cos4 $\Theta$ /16sin ² $\Theta$ ---------- (1)
Now, let us try to evaluate:
(m ²-n ²) ²
[cot ² $\Theta$ (1+sin ² $\Theta$ )/16- cot ² $\Theta$ (1-sin ² $\Theta$ )/16] ²
{[cot ² $\Theta$ (1+sin ² $\Theta$ ) - cot ² $\Theta$ (1-sin ² $\Theta$ )]/16} ²
[(4 sin $\Theta$ cot ² $\Theta$ )/16] ²
cos4 $\Theta$ /16sin ² $\Theta$ ----------------(2)
As equation (1) is equal to equation (2), so option 2 is the answer