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Answer:
[tex]\sec \theta + \csc \theta = \frac{2\cdot \sqrt{3}+6}{3}[/tex]
Step-by-step explanation:
We proceed to simplify the given trigonometric expression into a form with a single trigonometric function:
1) [tex]7\cdot \sin^{2}\theta + 3\cdot \cos^{2}\theta = 4[/tex] Given.
2) [tex]4\cdot \sin^{2}\theta + 3\cdot (\sin^{2}\theta + \cos^{2}\theta) = 4[/tex] Definition of addition/Associative and distributive properties.
3) [tex]4\cdot \sin^{2} \theta +3 = 4[/tex] [tex]\sin^{2}\theta + \cos^{2}\theta = 1[/tex]/Modulative property
4) [tex]4\cdot \sin^{2}\theta = 1[/tex] Compatibility with addition/Existence of additive inverse/Modulative property
5) [tex]\sin \theta = \frac{1}{2}[/tex] Compatibility with multiplication/Existence of multiplicative inverse/Modulative property/Definition of division
6) [tex]\theta = \sin^{-1} \frac{1}{2}[/tex] Inverse trigonometric inverse.
7) [tex]\theta = 30^{\circ}[/tex] Result.
By Trigonometry, we know that secant and cosecant functions have the following identities:
[tex]\sec \theta = \frac{1}{\cos \theta}[/tex], [tex]\csc \theta = \frac{1}{\sin \theta}[/tex] (1, 2)
In addition, we know that [tex]\sin 30^{\circ} = \frac{1}{2}[/tex] and [tex]\cos 30^{\circ} = \frac{\sqrt{3}}{2}[/tex], then the sum of the two trigonometric function abovementioned is:
[tex]\sec \theta + \csc \theta = \frac{1}{\cos \theta} + \frac{1}{\sin \theta}[/tex]
[tex]\frac{2}{\sqrt{3}} + 2 = \frac{2\cdot \sqrt{3}}{3} + 2[/tex]
[tex]\sec \theta + \csc \theta = \frac{2\cdot \sqrt{3}+6}{3}[/tex]