Show that tanx = sinx / cosx
can be written as
tan(x+y) = (tanx + tany) / (1 - tanxtany)
rewrite the left side as
sin(x+y)/cos(x+y), then use the addition formulas for both. It will simplify (finally) to the right.
To show that tanx = sinx / cosx can be written as tan(x + y) = (tanx + tany) / (1 - tanxtany), we need to use the trigonometric identities and properties. Here's how we can break it down step by step:
Step 1: Start with the given equation tanx = sinx / cosx.
Step 2: Express sinx and cosx in terms of tanx using the trigonometric identities. Recall that sinx = tanx * cosx and cosx = 1 / tanx.
Substituting these values, we get:
tan(x + y) = (tanx * cosy + tany) / (1 - tanx * tan(y + x))
Step 3: Simplify the expression using the distributive property:
tan(x + y) = (tanx * cosy + tany) / (1 - tanxtany)
Note that cosine of the sum of two angles (cos(x + y)) can be expressed as cos(x) * cos(y) - sin(x) * sin(y), which we can rewrite as cosy = cosx * cosy - sinx * siny.
Using this identity, we can further simplify the expression:
tan(x + y) = (tanx * (cosx * cosy - sinx * siny) + tany) / (1 - tanx * tany)
Step 4: Distribute the terms:
tan(x + y) = (tanx * cosx * cosy - tanx * sinx * siny + tany) / (1 - tanx * tany)
Step 5: Rearrange the terms:
tan(x + y) = (tanx * cosx * cosy + tany - tanx * sinx * siny) / (1 - tanx * tany)
Step 6: Rearrange the terms in the numerator:
tan(x + y) = (tanx * cosx * cosy - tanx * sinx * siny + tany) / (1 - tanx * tany)
Step 7: Factor out tanx from the numerator:
tan(x + y) = (tanx * (cosx * cosy - sinx * siny) + tany) / (1 - tanx * tany)
Step 8: Recognize that cosx * cosy - sinx * siny is equal to 1 by the Pythagorean identity: cos^2x + sin^2x = 1. This simplifies the equation to:
tan(x + y) = (tanx + tany) / (1 - tanx * tany)
Therefore, we have shown that tanx = sinx / cosx can be written as tan(x + y) = (tanx + tany) / (1 - tanxtany).