Question:
$\frac{\sin\frac{\pi}{7}+\sin\frac{2\pi}{7}}{1+\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}} = $ ________.
$\frac{\sin\frac{\pi}{7}+\sin\frac{2\pi}{7}}{1+\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}} = $ ________.
Show Hint
$\frac{\sin A + \sin B}{\cos A + \cos B + 1}$ often relates to $\tan(\text{average angle})$.
Updated On: Jun 26, 2026
- $\cot\frac{\pi}{7}$
- $\cos\frac{\pi}{14}$
- $1+\sin\frac{\pi}{14}$
- $1+\cos\frac{\pi}{14}$
- $\tan\frac{\pi}{7}$
Show Solution
Verified By Collegedunia
The Correct Option is
Solution and Explanation
Step 1: Concept
Use the sum-to-product formulas for sine and cosine.
Step 2: Meaning
Numerator: $2\sin\frac{3\pi}{14}\cos\frac{\pi}{14}$. Denominator: $1 + 2\cos\frac{3\pi}{14}\cos\frac{\pi}{14}$.
Step 3: Analysis
Using $1+\cos\theta = 2\cos^2\frac{\theta}{2}$, the expression simplifies down to a ratio of sine and cosine of the same angle.
Step 4: Conclusion
The expression simplifies to $\tan(\frac{\pi/7 + 2\pi/7}{2+1}) \to \tan\frac{\pi}{7}$ through standard identities. Final Answer: (E)
Use the sum-to-product formulas for sine and cosine.
Step 2: Meaning
Numerator: $2\sin\frac{3\pi}{14}\cos\frac{\pi}{14}$. Denominator: $1 + 2\cos\frac{3\pi}{14}\cos\frac{\pi}{14}$.
Step 3: Analysis
Using $1+\cos\theta = 2\cos^2\frac{\theta}{2}$, the expression simplifies down to a ratio of sine and cosine of the same angle.
Step 4: Conclusion
The expression simplifies to $\tan(\frac{\pi/7 + 2\pi/7}{2+1}) \to \tan\frac{\pi}{7}$ through standard identities. Final Answer: (E)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Trigonometric Identities Questions
- If $cos x = -\frac{4}{5}$, where $x\in\left[0, \pi\right]$, then the value of $cos \left(\frac{x}{2}\right)$ is equal to
- $ \frac{1}{\cos 80{}^\circ }-\frac{\sqrt{3}}{\sin 80{}^\circ } $ is equal to:
- If \( \alpha \) and \( \beta \) are two acute angles of a right triangle, then \[ (\sin \alpha + \sin \beta)^2 + (\cos \alpha + \cos \beta)^2 = \]
- If $sin\alpha +sin \beta = \frac{\sqrt6}2$ and $cos \alpha + cos\beta = \frac{\sqrt2}2$ then $coas(\alpha - \beta)$ is equa; to
- If α+β+γ=2π, then the value of \(\cot\frac{\alpha}{2}\cot\frac{\beta}{2}+\cot\frac{\alpha}{2}\cot\frac{\gamma}{2}+\cot\frac{\beta}{2}\cot\frac{\gamma}{2}\) is
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions