The given expression is:
[tex]\frac{\sin x\cot x\sec x}{\cos x\csc x}[/tex]By using trigonometric identities we have that:
[tex]\frac{\sin x}{\cos x}=\tan x[/tex]Then, replace sinx/cosx in the expression by tanx:
[tex]\frac{\sin x\cot x\sec x}{\cos x\csc x}=\frac{\tan x\cot x\sec x}{\csc x}[/tex]Now:
[tex]\tan x=\frac{1}{\cot x}[/tex]Thus:
[tex]\tan x\cdot\cot x=\frac{1}{\cot x}\cdot\cot x=\frac{\cot x}{\cot x}=1[/tex]Replace tanx*cotx in the expression by 1:
[tex]\frac{\tan x\cot x\sec x}{\csc x}=\frac{1\cdot\sec x}{\csc x}=\frac{\sec x}{\csc x}[/tex]Finally:
[tex]\begin{gathered} \sin x=\frac{1}{\csc x} \\ \text{and} \\ \frac{1}{\cos x}=\sec x \\ \text{Thus} \\ \frac{\sec x}{\csc x}=\frac{\sin x}{\cos x} \end{gathered}[/tex]And as we said in the first trigonometric identity sinx/cosx=tanx, thus:
[tex]\frac{\sin x}{\cos x}=\tan x[/tex]Answer: the trigonometric expression that is equal to the given is tanx
