How would you find tan80 and tan55 as needed in this problem?
Use identities to find each exact value.
(tan80+tan55)/(1-tan80tan55)
To find the exact values of tan80 and tan55, we can use trigonometric identities.
1. To find tan80:
We know that tan(180° - θ) = -tan(θ). Therefore, tan80 = -tan(180° - 80°) = -tan100.
2. To find tan55:
There is no direct trigonometric identity to find tan55. However, we can use the tangent addition formula to find an equivalent expression for tan55 in terms of known values.
The tangent addition formula states: tan(A + B) = (tanA + tanB) / (1 - tanA * tanB).
Let's use this formula with A = 45° and B = 10°.
tan55 = tan(45° + 10°) = (tan45 + tan10) / (1 - tan45 * tan10).
Since tan45 = 1 (known value), we can rewrite the expression as:
tan55 = (1 + tan10) / (1 - tan10)
Now we have the values of tan80 as -tan100 and tan55 as (1 + tan10) / (1 - tan10).
Using these values, we can substitute them into the given expression:
(tan80 + tan55) / (1 - tan80 * tan55)
= (-tan100 + (1 + tan10) / (1 - tan10)) / (1 - (-tan100) * ((1 + tan10) / (1 - tan10)))
Simplifying this expression will give you the exact value.