Answer:
A. [tex]-tan\theta[/tex]
Explanation:
Given
[tex]sin\theta\ .\ tan\theta\ .\ sec\theta .\ cot(-\theta)[/tex]
Required
Simplify
[tex]sin\theta\ .\ tan\theta\ .\ sec\theta .\ cot(-\theta)[/tex]
Substitute [tex]\frac{1}{cos\theta|}[/tex] for [tex]sec\theta[/tex]
So, the expression becomes
[tex]sin\theta\ .\ tan\theta\ .\ \frac{1}{cos\theta} .\ cot(-\theta)[/tex]
Rearrange the above expression
[tex]sin\theta\ .\ \frac{1}{cos\theta}\. \ tan\theta\ .\ cot(-\theta)[/tex]
[tex]\frac{sin\theta}{cos\theta}\. \ tan\theta\ .\ cot(-\theta)[/tex]
From trigonometry;
[tex]tan\theta = \frac{sin\theta}{cos\theta}[/tex]
So, we have
[tex]tan\theta\ . \ tan\theta\ .\ cot(-\theta)[/tex]
From trigonometry;
[tex]cot(-\theta) = -cot(\theta)[/tex]
So, the above expression becomes
[tex]tan\theta\ . \ tan\theta\ .\ -cot(\theta)[/tex]
[tex]-tan\theta\ . \ tan\theta\ .\ cot(\theta)[/tex]
From trigonometry;
[tex]cot\theta = \frac{1}{tan\theta}[/tex]
So, we have
[tex]-tan\theta\ . \ tan\theta\ .\ \frac{1}{tan\theta}[/tex]
Express as a single fraction
[tex]\frac{-tan\theta\ . \ tan\theta }{tan\theta}[/tex]
[tex]-tan\theta[/tex]
Hence, [tex]sin\theta\ .\ tan\theta\ .\ sec\theta .\ cot(-\theta) = -tan\theta[/tex]
