Question

(cos7pi/10 + sin7pi/10) / (cospi/5 + sinpi/5)

Answer 1

I hope this helps you

Answer 2

To simplify the expression $(\cos\frac{7\pi}{10} + \sin\frac{7\pi}{10}) / (\cos\frac{\pi}{5} + \sin\frac{\pi}{5})$, we can use trigonometric identities. Let's break it down step by step:

1. Use the angle sum formula for sine and cosine:

   $$\begin{align*} \cos\frac{7\pi}{10} &= \cos\left(\frac{\pi}{2} - \frac{3\pi}{10}\right) \\ &= \sin\frac{3\pi}{10} \end{align*}$$

   Similarly,

   $$\sin\frac{7\pi}{10} = \cos\frac{3\pi}{10}$$

   and

   $$\cos\frac{\pi}{5} = \cos\left(\frac{\pi}{2} - \frac{\pi}{10}\right) = \sin\frac{\pi}{10}$$

   and

   $$\sin\frac{\pi}{5} = \cos\frac{\pi}{10}$$

2. Substitute these values back into the original expression:

   $$\frac{\sin\frac{3\pi}{10} + \cos\frac{3\pi}{10}}{\sin\frac{\pi}{10} + \cos\frac{\pi}{10}}$$

3. Multiply the numerator and denominator by the conjugate of the denominator:

   $$\frac{(\sin\frac{3\pi}{10} + \cos\frac{3\pi}{10})(\sin\frac{\pi}{10} - \cos\frac{\pi}{10})}{(\sin\frac{\pi}{10} + \cos\frac{\pi}{10})(\sin\frac{\pi}{10} - \cos\frac{\pi}{10})}$$

4. Expand and simplify:

   $$\frac{\sin\frac{3\pi}{10}\sin\frac{\pi}{10} - \sin\frac{3\pi}{10}\cos\frac{\pi}{10} + \cos\frac{3\pi}{10}\sin\frac{\pi}{10} - \cos\frac{3\pi}{10}\cos\frac{\pi}{10}}{\sin^2\frac{\pi}{10} - \cos^2\frac{\pi}{10}}$$

5. Use trigonometric identities to simplify further:

   $$\frac{\sin\frac{4\pi}{10}}{\sin^2\frac{\pi}{10} - (1 - \sin^2\frac{\pi}{10})}$$

6. Simplify the denominator:

   $$\frac{\sin\frac{4\pi}{10}}{2\sin^2\frac{\pi}{10}}$$