Math Help - sum-product/product-sum

1. sum-product/product-sum

solve the following using sum-product/product-sum identities

$cos20+cos100+cos140=0$

$sin19cos11+sin71sin11=\frac{1}{2}$

$cos20cos40cos60cos180=\frac{1}{16}$

$cos20cos40cos80=\frac{1}{8}$

2. Re: sum-product/product-sum

In this case,

Q1,
$cos20+cos100+cos140=0$
L.H.S
= $2cos60cos40+cos140$

= $cos40+cos140$

= $2cos90cos50$

= $0$ (Since cos90=0)

=R.H.S

Q2,
$sin19cos11+sin71sin11=\frac{1}{2}$
L.H.S
= $\frac{2sin19cos11+2sin71sin11}{2}$

= $\frac{sin30+sin8+cos60-cos82}{2}$

= $\frac{\frac{1}{2}+\frac{1}{2}+sin(90-82)-cos82}{2}$

= $\frac{1+cos82-cos82}{2}$

= $\frac{1}{2}$

=R.H.S

Q3,
I think you mention cos180 or cos 80 I can't get any answer.

Q4,
$cos20cos40cos80=\frac{1}{8}$
L.H.S
= $\frac{2cos20cos40cos80}{2}$

= $\frac{(cos60+cos20)cos80}{2}$

= $\frac{(\frac{1+2cos20}{2})cos80}{2}$

= $\frac{\frac{cos80+cos100+cos60}{2}}{2}$

= $\frac{\frac{cos(180-100)+cos100+cos60}{2}}{2}$

= $\frac{\frac{-cos100+cos100+\frac{1}{2}}{2}}{2}$

= $\frac{1}{8}$

=R.H.S

3. Re: sum-product/product-sum

$\frac{2cos20cos40cos80}{2}$

Where did the 2 in the numerator and denominator come from???

4. Re: sum-product/product-sum

you want to multiply by 2 to numerator and denominator,This is a way to get your answer quickly.