Could someone please show me how
1 / ( 4 tan (x/2) + 2 )
is converted to:
(- sin (x/2) /( 2 sin (x/2) + cos (x/2))
TIA
Carol
lets check it at x=0, and x=PI/2
x=0: 1/4(1+2)=1/12
(-sin45/(2sin45+cos45)=-.707/(1.414+.707) does not equal 1/12
that was at x=90 deg, the identity did not hold. Now at x=0
1/4(0+2)=1/8
-0/(0+.7) does not equal 1/8
Sure, I'd be happy to explain how to convert the expression 1 / (4 tan(x/2) + 2) to (-sin(x/2) / (2 sin(x/2) + cos(x/2)).
To start, we'll tackle the denominator (4 tan(x/2) + 2). We need to rewrite this expression in terms of sine and cosine.
Using the identity tan(x) = sin(x) / cos(x), we can rewrite tan(x/2) as sin(x/2) / cos(x/2).
Now, let's substitute this into the denominator. We have:
4 * (sin(x/2) / cos(x/2)) + 2.
To simplify further, we can multiply through by cos(x/2) to get rid of the denominator:
4sin(x/2) + 2cos(x/2).
Now, let's move on to the numerator 1. We need to rewrite this in terms of sine and cosine as well.
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite 1 as cos^2(x/2) / cos^2(x/2):
cos^2(x/2) / cos^2(x/2).
Now, let's put everything together. We have:
(cos^2(x/2) / cos^2(x/2)) / (4sin(x/2) + 2cos(x/2)).
To simplify this, let's cancel out the common terms between the numerator and denominator. The cos^2(x/2) terms will cancel out, leaving us with:
1 / (4sin(x/2) + 2cos(x/2)).
Finally, we can further manipulate this expression by multiplying the numerator and denominator by -1. This gives us the desired expression:
-1 / -(4sin(x/2) + 2cos(x/2)).
Simplifying the negative sign, we get:
- sin(x/2) / (2sin(x/2) + cos(x/2)).
And there you have it! The expression 1 / (4 tan(x/2) + 2) is converted to -sin(x/2) / (2 sin(x/2) + cos(x/2)).