Evaluate: (5 cos 2 60 + 4 sec 2 30 – tan 2 45) / (sin 2 30 + sin 2 60)

Question

Q) Evaluate: $\frac{5 \cos ^2 60 + 4 \sec ^2 30 – 	an ^2 45} {\sin ^2 30 + \sin ^2 60}$

Sapling Academy · Accepted Answer

Let's take the components one by one:
Step 1: we know that cos 60 = $\frac{1}{2}$, sec 30 = $\frac{2}{\sqrt 3}$, tan 45 = 1,
sin 30 = $\frac{1}{2}$, sin 60 = $\frac{\sqrt 3}{2}$
Step 2: Let's put these values in the given expression, we get:
$\frac{5 \cos ^2 60 + 4 \sec ^2 30 – 	an ^2 45} {\sin ^2 30 + \sin ^2 60}$
∴ $\frac{5 (\frac{1}{2})^2 + 4 (\frac{2}{\sqrt 3})^2 – (1)^2} {(\frac{1}{2})^2  + (\frac{\sqrt 3}{2})^2}$
∴ $\frac{5 (\frac{1}{4}) + 4 (\frac{4}{3}) – (1)}{(\frac{1}{4})  + (\frac{3}{4})}$
∴ $\frac{(\frac{5}{4} + \frac{16}{3} – 1)}{(\frac{1}{4}  + \frac{3}{4})}$
∴ $\frac{(\frac{15 + 64 - 12}{12})}{(\frac{1 + 3}{4})}$
∴ $\frac{(\frac{67}{12})}{(\frac{4}{4})}$
∴ $\frac{67}{12}$
Therefore the value of given expression is $\frac{67}{12}$
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