Hello Johnny Walker Black Originally Posted by
Johnny Walker Black Verify these: [i need help getting started and other help you can throw at me]
1) sin(x + y)cos y - cos (x + y)sin y = sin x
2) cot x = (cos3x + cos x) / (sin 3x - sin x)
3) tan(B/2) = sec B / (sec B csc B + csc B)
Here's the method for each one:
1) Use the identities below to expand the LHS: $\displaystyle \sin(x+y) = \sin x \cos y +\cos x \sin y$
$\displaystyle \cos(x+y) = \cos x \cos y - \sin x \sin y$
Simplify the result. Then take out a common factor of $\displaystyle \sin x$. Then use:$\displaystyle \cos^2y + \sin^2y = 1$
and you're done.
2) Use the identities: $\displaystyle \cos 3x = 4\cos^3x - 3\cos x$
$\displaystyle \sin 3x = 3\sin x - 4\sin^3 x$
Simplify the result, and factorise. Then use:$\displaystyle \cos^2x = 1 - \sin^2 x$
in the numerator. Simplify; cancel, and you're there.
3) Get rid of $\displaystyle \sec B$ and $\displaystyle \csc B$ (horrible things!) by multiplying top-and-bottom of the fraction by $\displaystyle \sin B\cos B$, using the fact that: $\displaystyle \csc B\sin B = 1$ and $\displaystyle \sec B \cos B = 1$
Then use:$\displaystyle \sin B = 2 \sin\tfrac12B\cos\tfrac12B$
$\displaystyle \cos B = 2\cos^2\tfrac12B - 1$
Simplify. Then use $\displaystyle \tan\tfrac12B = \frac{\sin\tfrac12B}{\cos\tfrac12B}$, and you're there.
Grandad