Double angle identity. Just 5 questions. Really need help as my teacher has been frustrated with me

Question

asked Aug 1, 2020 88.7k views

2 Answers

VJune · Answer 1 · 2020-08-08T01:31:24+0000

The double angle identities are

$\sin(2x)=2\sin x\cos x$

$\cos(2x)=\cos^2x-\sin^2x$

Then

$2\sin\frac\pi6\cos\frac\pi6=\sin\left(\frac{2\pi}6\right)=\sin\frac\pi3$

$\cos^2\frac\pi{10}-\sin^2\frac\pi{10}=\cos\left((2\pi)/(10)\right)=\cos\frac\pi5$

The second identity together with the Pythagorean identity,
$\sin^2x+\cos^2x=1$ , gives us another equivalent expression:

$\cos^2x-\sin^2x=\cos^2x-(1-\cos^2x)=2\cos^2x-1$

so

$2\cos^2(0.5)-1=\cos(2\cdot0.5)=\cos1$