Final answer:
The identity (sinx+sin2x)/(1+cosx+sin2x)=tan x is proven by using trigonometric identities to factor and simplify both the numerator and the denominator.
Step-by-step explanation:
To prove that (sinx+sin2x)/(1+cosx+sin2x)=tan x, we will use trigonometric identities to transform the left side of the equation into the right side.
First, let us observe the double angle identities: sin2x = 2sinx cosx and cos2x = cos2x - sin2x which can also be written as 1 - 2sin2x.
Now, let's substitute sin2x in the numerator of the left side of the equation with 2sinx cosx which simplifies down to:
This expression can be factored as:
The denominator can be left as is, since 1 + cosx + sin2x already includes the double angle for sin.
If we notice that tan x = sin x / cos x, let us now make cos x the subject in the denominator by rewriting 1 as cos2x + sin2x (from the Pythagorean identity), yielding:
- cos2x + sin2x + 2sinx cosx
Which is equal to:
- (cosx + sinx)(1 + 2 cosx)
Therefore, after factoring the numerator and rearranging the denominator:
- (sinx(1 + 2 cosx))/(cosx(1 + 2 cosx)) = sinx/cosx
sinx/cosx is equal to tan x, proving our initial equation.