Question

verify the identities:
1. x=60 degrees, sin 2x=2 sin x cos x
2. cos (2theta)/sin theta cos theta = cot theta minus tan theta

Answer 1

in (1), just substitute x=60 deg to the given identity of sin 2x...

Answer 2

sin 2x = 2 sin x cos x is already an identity derived as double angle formula... using
sin (x + x) = sin x cos x + sin x cos x = 2 sin x cos x

Answer 3

in (2) we use double angle formula for cos 2theta...
\[\cos 2\theta=\cos^2\theta-\sin^2\theta\]

Answer 4

\[\frac{ \cos2\theta }{ \sin \theta \cos \theta }=\cot \theta-\tan \theta\]
\[\frac{ \cos^2\theta }{ \sin \theta \cos \theta }-\frac{ \sin^2\theta }{ \sin \theta \cos \theta }=\cot \theta-\tan \theta\]
\[\frac{ \cos \theta }{ \sin \theta }-\frac{ \sin \theta }{ \cos \theta }=\cot \theta-\tan \theta\]
that proves the identity for (2)...

Answer 5

since \[\cot \theta = \frac{ \cos \theta }{ \sin \theta }\] and \[\tan \theta = \frac{ \sin \theta }{ \cos \theta }\]