Question

Someone please help me to prove this. ​

Answer 1

Answer: see proof below

Explanation:

Use the following Product to Sum Identities:

2 sin A · cos B = sin (A + B) + sin (A - B)

2 cos A · cos B = sin (A + B) + sin (A - B)

Given: cos A + cos B = 1/2 and sin A + sin B = 1/4

Proof LHS → RHS

`\text{LHS:}\qquad \qquad \qquad \tan(A+B)/(2)`

`\text{Expand:}\qquad \qquad (\sin((A+B))/(2))/(\cos((A+B))/(2))`

`\text{Multiplication:}\qquad \quad (\sin((A+B))/(2))/(\cos((A+B))/(2))\bigg((2\cos(A-B)/(2))/(2\cos (A-B)/(2))\bigg)`

`\text{Simplify:}\qquad \qquad \quad (2\sin (A+B)/(2)\cdot \cos (A-B)/(2))/(2\cos (A+B)/(2)\cdot \cos (A-B)/(2))`

`\text{Product to Sum:}\qquad (\sin A+\sin B)/(\cos A+\cos B)`

`\text{Given:}\qquad \qquad \qquad \quad ((1)/(4))/((1)/(2))`

`\text{Simplify:}\qquad \qquad \qquad (1)/(2)`

LHS = RHS: 1/2 = 1/2
`\checkmark`