You can't prove a function. You can prove an identity.
Using s for sinx, c for cosx, t for tanx, to make things look less cumbersome:
(s+2sc)/(1+c+2c^2 - 1) = t
(s+2sc)/(c + 2c^2) = t
s(1+2c)/[c(1+2c)] = t
s/c = t
t = t
(sinx+sin2x)/(1+cosx+cos2x)=tan x
(cos3x/sinx)+(sin3x/cosx)=2cot2x
tan2x=(2/cotx-tanx)
theses are due in the am. Fastness would be great?
Using s for sinx, c for cosx, t for tanx, to make things look less cumbersome:
(s+2sc)/(1+c+2c^2 - 1) = t
(s+2sc)/(c + 2c^2) = t
s(1+2c)/[c(1+2c)] = t
s/c = t
t = t
1. (sinx + sin2x) / (1 + cosx + cos2x) = tanx
To prove this function, we will convert everything to sines and cosines.
First, let's express the numerator and denominator of the left side in terms of sines and cosines:
sinx + sin2x = sinx + 2sinx*cosx
1 + cosx + cos2x = 1 + cosx + 2cos^2(x) - 1 = 2cos^2(x) + cosx
Now, rewrite the expression using trigonometric identities:
(sin x + 2sin x * cos x) / (2cos x * cos x + cos x) = tan x
To simplify further, cancel out the common terms:
sin x / cos x = tan x
tan x = tan x
Thus, we have proved the given function.
2. (cos3x / sin x) + (sin3x / cos x) = 2cot2x
Let's start by rewriting the expression:
(cos 3x / sin x) + (sin 3x / cos x) = (cos 3x * cos x + sin 3x * sin x) / (sin x * cos x)
Using the trigonometric identity cos(A + B) = cos A * cos B - sin A * sin B, we can simplify further:
(cos (3x - x)) / (sin x * cos x) = cos 2x / (sin x * cos x)
Next, we can use the identity cot A = cos A / sin A:
cos 2x / (sin x * cos x) = 2cos 2x / (2sin x * cos x) = 2cot 2x
Hence, we have proved the given function.
3. tan 2x = (2 / cot x - tan x)
First, let's substitute the values of cot x and tan x:
tan 2x = (2 / (cos x / sin x) - (sin x / cos x))
Next, multiply the numerator and denominator by sin x * cos x to rationalize the expression:
tan 2x = (2sin x * cos x) / (cos x - sin^2 x) - (sin^2 x)
Using the identity 1 - sin^2 x = cos^2 x, we can simplify it further:
tan 2x = (2sin x * cos x) / cos^2 x = (2sin x) / cos x = 2tan x
Therefore, we have proved the given function.
I hope the explanation is clear and helpful. If you have any further questions or need more clarification, please let me know.