[spec] corrections in double angle formulas

diff --git a/spec/circ.md b/spec/circ.md
index c3c5b173675e8db11d22477f347d86d814786223..a5dae02834d3dc1329cfddcf5d76fbdd32cb48c8 100644 (file)
--- a/spec/circ.md
+++ b/spec/circ.md
@@ -77,7 +77,9 @@ $$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
 ## Double angle formulas
 
 \begin{equation}\begin{split}
 ## Double angle formulas
 
 \begin{equation}\begin{split}

-  \cos 2x = \cos^2 x = \sin^2 x
+  \cos 2x &= \cos^2 x - \sin^2 x \\
+  & = 1 - 2\sin^2 x \\
+  & = 2 \cos^2 x -1

 \end{split}\end{equation}
 
 $$\sin 2x = 2 \sin x \cos x$$
 \end{split}\end{equation}
 
 $$\sin 2x = 2 \sin x \cos x$$

diff --git a/spec/circ.pdf b/spec/circ.pdf
index 4f0cd6684892577815600400a32e5cd5a1a10076..bb10a7f885ffb7257115d76d3bcedaafa262ae9f 100644 (file)
Binary files a/spec/circ.pdf and b/spec/circ.pdf differ