Expanding double angles and simplifying trigonometry

Question

guys I am struggling with the expansion from the second last line to the last for this problem. Any suggestions with examples would be highly appreciated. Thanks a lot

Answer 1

You use $\cos(A+B)=\cos A\cos B-\sin A\sin B,\cos(A-B)=\cos A\cos B+\sin A\sin B$. So $\cos(A-B)-\cos(A+B)=2\sin A\sin B$.

Takng $A=5\pi10^7t-0.2\pi,B=0.2\pi$ we get $\cos(5\pi10^7t-0.4\pi)-\cos(5\pi10^7t)=2\sin(5\pi10^7t-0.2\pi)\sin(0.2\pi)$

So we have the desired expression provided $3\sin(0.2\pi)=1.76$.

That is indeed true - either use a calculator or note that $\sin\frac{\pi}{5}=\sqrt{\frac{5-\sqrt5}{8}}$

Answer 2

$$\cos(C)-\cos(D)=2\sin({C+D\over 2})\sin({D-C\over 2})$$

Take $C=5\pi 10^7t-0.4\pi$ and $D=5\pi10^7t$

Now use a calculator