Question

Cos(70+x)*cos(x-20) = 1/2

Answer 1

1/2*(сos90+сos(2x+50))=1/2
cos(2x+50)=1
2x+50=180n
2x=-50+180n
x=25+90n,n∈z

Answer 2

We can use the trigonometric identity:

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

to simplify the left-hand side of the equation. Applying this identity, we get:

cos(70+x)*cos(x-20) = [cos(70)cos(x) - sin(70)sin(x)][cos(x)cos(20) + sin(x)sin(20)]

= cos(70)cos(x)cos(20) + sin(70)sin(x)sin(20) - sin(70)cos(x)sin(20) + sin(x)cos(70)cos(20)

= [cos(70)cos(20) - sin(70)sin(20)]cos(x) + [sin(70)cos(20) + cos(70)sin(20)]sin(x)

= cos(50)cos(x) + sin(50)sin(x)

= cos(x-50)

Therefore, we have:

cos(x-50) = 1/2

Taking the inverse cosine of both sides, we get:

x - 50 = ±π/3 + 2πk, where k is an integer

Solving for x, we get:

x = 50 ±π/3 + 2πk, where k is an integer

So the general solution to the equation is:

x = 50 + π/3 + 2πk or x = 50 - π/3 + 2πk, where k is an integer.