[tex]\displaystyle\\
\alpha = 37^o \\
\beta = 53^o \\
\alpha+\beta=37^o+53^o=90^o \\
\Longrightarrow~~\alpha \text{ and } \beta \text{ are complementary.} \\ \\
\csc (37^o) \sec(53^o) - \tan (53^o) \cot (37^o)= \\ \\
=\csc (90^o-53^o) \sec(53^o) - \tan (53^o) \cot (90-53^o)= \\ \\
=\sec (53^o) \sec(53^o) - \tan (53^o) \tan (53^o)= \\ \\
=\sec^2 (53^o) - \tan^2 (53^o)= \\ \\
= \frac{1}{\cos^2(53^o)} - \tan^2 (53^o)=\\\\
= \frac{\sin^2(53^o)+\cos^2(53^o)}{\cos^2(53^o)} - \tan^2 (53^o)= [/tex]
[tex]\displaystyle\\
= \frac{\sin^2(53^o)}{\cos^2(53^o)} +\frac{\cos^2(53^o)}{\cos^2(53^o)} - \tan^2 (53^o)= \\ \\
= \frac{\sin^2(53^o)}{\cos^2(53^o)} +1- \tan^2 (53^o)= \\ \\
=\underline{\tan^2 (53^o)}+1- \underline{\tan^2 (53^o)}= \boxed{\bold{1}}[/tex]
