Double-Angle Formulas

$\blacktriangleright$ Expand

sin 2θ =  2 sinθcosθ 4pt

cos 2θ =  2 cos2θ - 1

tan 2θ =  2$\left(\vphantom{ \sin \theta }\right.$sinθ$\left.\vphantom{ \sin \theta }\right)$${\dfrac{{\cos \theta }}{{2\cos
^{2}\theta -1}}}$ 8pt

You can uncover other multiple-angle formulas with Expand. Following are some examples.

$\blacktriangleright$ Expand

sin 6θ =  32 sinθcos5θ -32 sinθcos3θ +6 sinθcosθ

sin 24θ = 8388608 sinθcos23θ  -  46137344 sinθcos21θ

                         +  110100480 sinθcos19θ  -  149422080 sinθcos17θ

                         +  127008768 sinθcos15θ  -  70189056 sinθcos13θ

                         +  25346048 sinθcos11θ  -  5857280 sinθcos9θ

                         +  823680 sinθcos7θ  -  64064 sinθcos5θ

                         +  2288 sinθcos3θ  -  24 sinθcosθ

sin(2a + 3b) =  8 sin a cos a cos3b  -  6 sin a cos a cos b

                                 +   8 cos2a sin b cos2b  -  2 cos2a sin b

                                 -   4 sin b cos2b  +  sin b