Answer:
[tex]\huge\boxed{ \bf\:1}[/tex]
Step-by-step explanation:
The key element to solve this question is to know the trignometric values of the given angles.
cosec θ, sec θ & cot θ are the reciprocals of sin θ, cos θ & tan θ respectively.
Please refer to the attachment for the trignometric values of 30°, 45° & 60° angles as they are used in the given question.
[tex]\rule{150}{2}[/tex]
Now, let's solve this question.
First, let's write the values of the given trignometric degrees.
[tex]\star\sec^{2}(60) = 2^{2}= 4\\ \star\tan^{2}(60) = (\sqrt{3} )^{2} = 3\\\star\sin^{2}(30) = (\frac{1}{2} )^{2} = \frac{1}{4} \\\star\cos^{2}(30) =( \frac{\sqrt{3} }{2} )^{2} = \frac{3}{4}[/tex]
Now, let's solve the given question by substituting the above values & then simplifying by doing the necessary arithmetic operations.
[tex]\sf\:\frac{\sec^{2}(60) - \tan^{2}(60)}{\sin^{2}(30)+\cos^{2}(30)}\\\sf\:= 4-3 \: \div \frac{1}{4} + \frac{3}{4} \\\sf\:= 1 \div \frac{4}{4} \\\sf\:= \frac{1}{1} * \frac{4}{4} (reciprocal) \\\sf\:= \frac{4}{4} \\=\boxed{ \bf\:1}[/tex]
[tex]\rule{150}{2}[/tex]
