Hi
Can someone explain the following diagram I have drawn.
For example where the t and 2 came from and their use in identities as I cannot make the relationship between them in trig identities involving tan
Thanks
Using pythagorean theorem you can see the relationship between the sides.
$\displaystyle A^2 = B^2 + C^2$
Where A is the hypotenuse (across from the 90 degree angle) and B and C are the vertical and horizontal sides.
So,
$\displaystyle A^2 = (2t)^2 + (1-t^2)^2$
$\displaystyle A^2 = 4t^2 + (1-2t^2+t^4) = t^4 +2t^2 + 1$
The right side is a perfect square and thus simplifies to:
$\displaystyle A^2 = (t^2 + 1)^2$
Rooting both side we see A, the hypotenuse, is $\displaystyle t^2+1$
Does this clarify things for you?
Hi
$\displaystyle \sin 2t = 2 \sin t \cos t$
$\displaystyle \cos 2t = \cos^2 t - \sin^2 t$
Therefore $\displaystyle \tan 2t = \frac{2 \sin t \cos t}{\cos^2 t - \sin^2 t}$
Dividing numerator and denominator by $\displaystyle \cos^2 t$
$\displaystyle \tan 2t = \frac{2 \tan t}{1 - \tan^2 t}$
Let T = tan(t)
$\displaystyle \tan 2t = \frac{2T}{1 - T^2}$