Prove that Sin x + tan x / 1+cosx =tan x
To prove that the expression (sin x + tan x) / (1 + cos x) is equal to tan x, we can manipulate each side of the equation separately until they match.
We will first work on the left side of the equation:
(sin x + tan x) / (1 + cos x)
Using the identity sin x = (1 - cos²x)^(1/2), we can rewrite the numerator as:
sin x = (1 - cos²x)^(1/2)
Now let's substitute these values back into the equation:
((1 - cos²x)^(1/2) + tan x) / (1 + cos x)
Next, let's simplify the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (1 - cos x):
(((1 - cos²x)^(1/2) + tan x) * (1 - cos x)) / ((1 + cos x) * (1 - cos x))
Expanding the numerator:
((1 - cos²x)^(1/2) - cos x + tan x - tan x * cos x) / (1 - cos²x)
Next, distribute the denominator:
((1 - cos²x)^(1/2) - cos x + tan x - tan x * cos x) / (1 - cos²x)
Now, let's simplify the numerator:
((1 - cos²x)^(1/2) - cos x + tan x - tan x * cos x) / (1 - cos²x)
= ((1 - cos²x)^(1/2) - cos x + tan x * (1 - cos x)) / (1 - cos²x)
Next, we can use the identity tan x = sin x / cos x:
(tan x * cos x) = sin x
Substituting this back into the equation:
((1 - cos²x)^(1/2) - cos x + sin x) / (1 - cos²x)
Now, let's use the identity (1 - cos²x) = sin²x:
(1 - cos²x)^(1/2) = sin x
Substituting this back into the equation:
(sin x - cos x + sin x) / (1 - cos²x)
= (2sin x - cos x) / (1 - cos²x)
Finally, we can use the identity (1 - cos²x) = sin²x one more time:
(2sin x - cos x) / (1 - cos²x)
= (2sin x - cos x) / sin²x
Now, we can simplify the right side of the equation:
tan x = sin x / cos x
= (2sin x - cos x) / sin²x
As we can see, the left side of the equation (sin x + tan x) / (1 + cos x) has been simplified to (2sin x - cos x) / sin²x, which is equivalent to the right side of the equation tan x.
Therefore, we have proven that (sin x + tan x) / (1 + cos x) = tan x.