Evaluate the following expression by substituting the standard trigonometric values: $$\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - an^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$
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Q: Evaluate the following expression by substituting the standard trigonometric values: $$\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$\nShow all steps clearly.

Evaluation of the Trigonometric Expression \nGiven expression: $$\text{Expression} = \frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$ Step 1: Recall the standard trigonometric values for $30^\circ$, $45^\circ$, and $60^\circ$: $\cos 60^\circ = \frac{1}{2}$ $\sec 30^\circ = \frac{2}{\sqrt{3}}$ $\tan 45^\circ = 1$ $\sin 30^\circ = \frac{1}{2}$ $\cos 30^\circ = \frac{\sqrt{3}}{2}$ Step 2: Substitute these values into the numerator and denominator. Numerator calculation: $$5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ = 5 \left(\frac{1}{2}\right)^2 + 4 \left(\frac{2}{\sqrt{3}}\right)^2 - (1)^2$$ Now, compute the squares: $$\left(\frac{1}{2}\right)^2 = \frac{1}{4}$$ $$\left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}$$ $$(1)^2 = 1$$ Substitute the squared values back into the numerator: $$\text{Numerator} = 5 \left(\frac{1}{4}\right) + 4 \left(\frac{4}{3}\right) - 1$$ $$\text{Numerator} = \frac{5}{4} + \frac{16}{3} - 1$$ Find the common denominator for the numerator (LCM of $4$ and $3$ is $12$): $$\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}$$ $$\frac{16}{3} = \frac{16 \times 4}{3 \times 4} = \frac{64}{12}$$ $$1 = \frac{12}{12}$$ Combine the numerator terms: $$\text{Numerator} = \frac{15 + 64 - 12}{12} = \frac{79 - 12}{12} = \frac{67}{12}$$ Denominator calculation: $$\sin^2 30^\circ + \cos^2 30^\circ = \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2$$ $$\text{Denominator} = \frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1$$ (Note: This also directly follows from the fundamental trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ for $\theta = 30^\circ$.) Step 3: Divide the simplified numerator by the denominator. $$\text{Final Value} = \frac{\frac{67}{12}}{1} = \frac{67}{12}$$ Final Answer:\nThe value of the given expression is $\frac{67}{12}$.

Question

Evaluate the following expression by substituting the standard trigonometric values: $$\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - 	an^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$
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Accepted Answer

Evaluation of the Trigonometric Expression 
Given expression: $$	ext{Expression} = \frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - 	an^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$ Step 1: Recall the standard trigonometric values for $30^\circ$, $45^\circ$, and $60^\circ$: $\cos 60^\circ = \frac{1}{2}$ $\sec 30^\circ = \frac{2}{\sqrt{3}}$ $	an 45^\circ = 1$ $\sin 30^\circ = \frac{1}{2}$ $\cos 30^\circ = \frac{\sqrt{3}}{2}$ Step 2: Substitute these values into the numerator and denominator. Numerator calculation: $$5 \cos^2 60^\circ + 4 \sec^2 30^\circ - 	an^2 45^\circ = 5 \left(\frac{1}{2}ight)^2 + 4 \left(\frac{2}{\sqrt{3}}ight)^2 - (1)^2$$ Now, compute the squares: $$\left(\frac{1}{2}ight)^2 = \frac{1}{4}$$ $$\left(\frac{2}{\sqrt{3}}ight)^2 = \frac{4}{3}$$ $$(1)^2 = 1$$ Substitute the squared values back into the numerator: $$	ext{Numerator} = 5 \left(\frac{1}{4}ight) + 4 \left(\frac{4}{3}ight) - 1$$ $$	ext{Numerator} = \frac{5}{4} + \frac{16}{3} - 1$$ Find the common denominator for the numerator (LCM of $4$ and $3$ is $12$): $$\frac{5}{4} = \frac{5 	imes 3}{4 	imes 3} = \frac{15}{12}$$ $$\frac{16}{3} = \frac{16 	imes 4}{3 	imes 4} = \frac{64}{12}$$ $$1 = \frac{12}{12}$$ Combine the numerator terms: $$	ext{Numerator} = \frac{15 + 64 - 12}{12} = \frac{79 - 12}{12} = \frac{67}{12}$$ Denominator calculation: $$\sin^2 30^\circ + \cos^2 30^\circ = \left(\frac{1}{2}ight)^2 + \left(\frac{\sqrt{3}}{2}ight)^2$$ $$	ext{Denominator} = \frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1$$ (Note: This also directly follows from the fundamental trigonometric identity $\sin^2 	heta + \cos^2 	heta = 1$ for $	heta = 30^\circ$.) Step 3: Divide the simplified numerator by the denominator. $$	ext{Final Value} = \frac{\frac{67}{12}}{1} = \frac{67}{12}$$ Final Answer:
The value of the given expression is $\frac{67}{12}$.

CBSE · Class 10 · Mathematics · Introduction to TrigonometryEvaluate the following expression by substituting the standard trigonometric values: $$\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$\nShow all steps clearly.

Evaluation of the Trigonometric Expression