Prove the following functions:
(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?
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
tanx
To prove the given functions, we need to manipulate the expressions on both sides of the equation and simplify them until they are in the same form. Let's start with each function one by one:
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.