We can simplify the expression step by step:
[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} ]
[ \tan(80° + 55°) = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
[ \tan 135° = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
[ -1 = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
[ \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} + 1 = 0 ]
So the final expression simplifies to 0.
We can simplify the expression step by step:
Recall the trigonometric identity:[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} ]
Let A = 80° and B = 55°, then:[ \tan(80° + 55°) = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
Substitute the values:[ \tan 135° = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
Simplify the expression:[ -1 = \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} ]
Add 1 to both sides:[ \frac{\tan 80° + \tan 55°}{1 - \tan 80° \tan 55°} + 1 = 0 ]
So the final expression simplifies to 0.