tan(X+30)tan(30-x)=2cos2x-1 / (2cos2x+1)
i just did
(tanx)(-tanx)=-sin^2x/cos^2x would that work?
no, just test it for some value of x
LS is not equal to your RS
Carry, where are you getting these,
even though I enjoy doing them, they are getting to me, lol
recall tan(A+B) = (tanA + tanB)/(1 - tanAtanB)
so LS
= (tanx + tan30)/1-tanxtan30)*(tan30-tanx)/(1+tanxtan30)
= (tan^2 30 - tan^2 x)/(1 - (tan^x)(tan^2 30))
now tan^2 30º = 1/3
so the above
= (1/3 - tan^2 x)/(1 - (1/3)tan^2 x)
multiply top and bottom by 3 to get
(1 - 3tan^2 x)/(3 - tan^2 x)
RS = (2cos(2x) - 1)/(2cos(2x) + 1)
= (2(cos^2x - sin^2x) - 1)/(2(cos^2x-sin^2x) + 1)
remember 1 = sin^2x + cos^2x , so we get
(2cos^2x - 2sin^x - sin^2x - cos^2x)/(2cos^2x - 2sin^2x + sin^2x + cos^2x)
= (cos^2x - 3sin^2x)/(3cos^2x - sin^2x)
divide top and bottom by cos^x to get
(1 - 3tan^2x)/(3 - tan^2x)
ok, no more!!!!
To verify if your simplification is correct, let's expand both sides of the given equation and see if they are equal.
Starting with the left-hand side (LHS):
LHS = tan(x+30)tan(30-x)
Using the tangent addition formula, we can rewrite the above expression:
LHS = [(tan(x) + tan(30))/(1 - tan(x)tan(30))][(tan(30) - tan(x))/(1 + tan(x)tan(30))]
Next, simplifying the expression above:
LHS = [(tan(x) + √(3))/(1 - tan(x)√(3))][(-√(3) - tan(x))/(1 + tan(x)√(3))]
Now, multiplying the two fractions:
LHS = [(-√(3)tan(x) - tan^2(x) - √(3)tan(x) - 3)/(1 - 3tan^2(x))]/[1 - √(3)tan(x) + √(3)tan(x) - 3tan^2(x)]
LHS = (-2√(3)tan(x) - tan^2(x) - 3)/(1 - 3tan^2(x))
Moving on to the right-hand side (RHS):
RHS = (2cos(2x) - 1)/(2cos(2x) + 1)
Using the double-angle formula for cosine, we can rewrite the above expression:
RHS = (2(2cos^2(x) - 1) - 1)/(2(2cos^2(x) - 1) + 1)
Simplifying:
RHS = (4cos^2(x) - 2 - 1)/(4cos^2(x) - 2 + 1)
RHS = (4cos^2(x) - 3)/(4cos^2(x) - 1)
Both sides have been simplified. Now, let's compare them:
LHS = (-2√(3)tan(x) - tan^2(x) - 3)/(1 - 3tan^2(x))
RHS = (4cos^2(x) - 3)/(4cos^2(x) - 1)
As you can see, the left-hand side (LHS) and right-hand side (RHS) are not the same. Therefore, your initial simplification is not correct.
It's always important to double-check your simplifications and verify whether both sides of the equation are indeed equal.