Question

Please look at the image this is a homework problem I would appreciate help with!

Answer 1

`\sin (t+2\pi)-\cos (t+4\pi)+\tan (t+\pi)=c`

Explanation:

We have the following expression:

`\sin (t+2\pi)-\cos (t+4\pi)+\tan (t+\pi)`

Now, we apply the angle sum identities in each case, just like this:

`\begin{gathered} \sin (t+2\pi)=\sin t\cdot\sin 2\pi+\cos t\cdot\cos 2\pi \\ \cos (t+4\pi)=\cos t\cdot\cos 4\pi+\sin t\cdot\sin 4\pi \\ \tan (t+\pi)=(\tan t+\tan\pi)/(1+\tan t\cdot\tan\pi) \end{gathered}`

We replace by the equivalences of a, b and c, like this:

`\begin{gathered} \sin t\cdot\sin 2\pi+\cos t\cdot\cos 2\pi=a\cdot0+b\cdot1=b \\ \cos t\cdot\cos 4\pi+\sin t\cdot\sin 4\pi=b\cdot1+a\cdot0=b \\ (\tan t+\tan\pi)/(1+\tan t\cdot\tan\pi)=(c+0)/(1+c\cdot0)=(c)/(1)=c \\ \\ \text{ Replacing:} \\ b-b+c=c \end{gathered}`