Question

Evaluate :

$\sf{\cfrac{5{Cos}^{2}60°+4{Cos}^{2}30°-{Tan}^{2}45°}{{Sin}^{2}30°+{Cos}^{2}30°}}$​

Answer 1

Given :

• $\sf{\cfrac{5{Cos}^{2}60°+4{Cos}^{2}30°-{Tan}^{2}45°}{{Sin}^{2}30°+{Cos}^{2}30°}}$

According to the question :

$\\\implies\bf\sf{\frac{5{Cos}^{2}60°+4{Cos}^{2}30°-{Tan}^{2}45°}{{Sin}^{2}30°+{Cos}^{2}30°}}$

$\\\implies\bf\sf{Sin^{2}30°+ {Cos^{2}30°} = {Sin^{2}\:θ+Cos^{2}\:θ = 1}}$

$\\\implies\bf\sf{cos\:60°\:=\:{ \frac{1}{2}}}\:and\:{sec\:60°\:=\:{ \frac{2}{ \sqrt{3}}}}$

Substituting those values,

$\\\implies\bf\sf{ \frac{5\:×\:(1/2)^{2}\:+\:4\:×\:(2/√3)^{2}\:-\:12}{1}}$

$\\\implies\bf\sf{ \frac{5}{4}}\:+\:4\:({ \frac{4}{3}})\:-\:1$

$\\\implies\bf\sf{ \frac{5}{4}}\:+\:4\:({ \frac{16}{3}})\:-\:1$

$\\\implies\bf\sf{ \frac{15\:+\:64\:-\:12}{12}}$

$\\\implies\bf\sf{ \frac{67}{12}}$

$\\\implies\bf\sf5\:{ \frac{7}{12}}$

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So Its Done !!