By expressing the trigonometric functions in terms of sine and cosine, the simplified expression is 1 / (cos α * (cot β + tan α)).
How to simplify the expression
To simplify the expression (tan α + tan β) / (cot α + cot β), start by expressing the trigonometric functions in terms of sine and cosine:
tan α = sin α / cos α
tan β = sin β / cos β
cot α = cos α / sin α
cot β = cos β / sin β
Substitute these expressions into the original expression:
(tan α + tan β) / (cot α + cot β) = (sin α / cos α + sin β / cos β) / (cos α / sin α + cos β / sin β)
Find a common denominator for the numerators and denominators:
[(sin α * sin β) / (cos α * cos β)] / [(cos α * sin β + cos β * sin α) / (sin α * sin β)]
Simplify further by canceling out sin α * sin β:
1 / [(cos α * cos β) / (cos α * sin β + cos β * sin α)]
Now, simplify the denominator by factoring out a common factor of cos α:
1 / [cos α * (cos β / sin β + sin α)]
Finally, rewrite the expression as:
1 / (cos α * (cot β + tan α))
Therefore, the simplified expression is 1 / (cos α * (cot β + tan α)).