Question

P+Q+R=180 then to prove (cosp/sinq.sinr) + (cosq/sinr.sinp) + (cosr/sinp.sinq)=2

Answer 1

Follows are the solution to this question:

Explanation:

Given:

`\to P+Q+R=180`

prove that
`((\cos p)/(\sin q \cdot \sin r)) +((\cos q)/(\sin r \cdot \sin p))+ ((\cos r)/(\sin p \cdot \sin q))=2`

Solving the L.H.S part:

`\to ((\cos p)/(\sin q \cdot \sin r)) +((\cos q)/(\sin r \cdot \sin p))+ ((\cos r)/(\sin p \cdot \sin q))\\\\`

Solve the above value by taking L.C.M:

`\to ((\cos p \sin p+\cos q \sin q +\cos r \sin r)/( \sin p \cdot \sin q \cdot \sin r)) \\\\\to ((\cos p \sin p+\cos q \sin q +\cos r \sin r)/( \sin p \cdot \sin q \cdot \sin r)) \\\\`

multiply the above value by
`(2)/(2)`:

`\to ((\cos p \sin p+\cos q \sin q +\cos r \sin r)/( \sin p \cdot \sin q \cdot \sin r)) * (2)/(2) \\\\\to (2 \cos p \sin p+ 2 \cos q \sin q + 2 \cos r \sin r)/(2 \sin p \cdot \sin q \cdot \sin r) \\\\\therefore \sin 2A = 2 \sin A \cdot \cos A\\\\\because\\\\\to ( \sin 2p+ \sin 2q + \sin 2r)/(2 \sin p \cdot \sin q \cdot \sin r) \\\\\therefore \ \\ \bold{\sin 2p+ \sin 2q + \sin 2r= 4 \sin p \cdot \sin q \cdot \sin r}\\`

`\to (4 \sin p \cdot \sin q \cdot \sin r)/(2 \sin p \cdot \sin q \cdot \sin r) \\\\\to (2 \sin p \cdot \sin q \cdot \sin r)/( \sin p \cdot \sin q \cdot \sin r) \\\\\to 2`

So, L.H.S=R.H.S