Prove sin(a+ beta)/cos alpha cos beta = tan alpha+tan beta

Answer 1

Using the angle addition identity,

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) so we can re-write the problem:

sin(α + β)/[cos(α)cos(β)]

[sin(α)cos(β) + cos(α)sin(β)]/[cos(α)cos(β)]

Now, we can split this "fraction" apart into it's two pieces:

[sin(α)cos(β)]/[cos(α)cos(β)] + [cos(α)sin(β)]/[cos(α)cos(β)]

Now cancel cos(β) in the first term and cos(α) in the right term:

sin(α)/cos(α) + sin(β)/cos(β)

Using the identity tan(x) = sin(x)/cos(x), we can re-write this as:

tan(α) + tan(β)