Answer:
Therefore, 1 + 2cos a can be expressed as the product of 2 and (cos a/2 + 1).
1 + 2cos a = 2(cos a/2 + 1)
Explanation:
We can use the trigonometric identity:
cos 2a = 1 - 2 sin^2 a
to rewrite 1 + 2cos a as:
1 + 2cos a = 1 + 2(1 - sin^2 a/2)
= 1 + 2 - 2(sin^2 a/2)
= 3 - 2(sin^2 a/2)
Now, using another trigonometric identity:
sin a = 2 sin(a/2) cos(a/2)
we can rewrite sin^2 a/2 as:
sin^2 a/2 = (1 - cos a)/2
Substituting this into the expression for 1 + 2cos a, we get:
1 + 2cos a = 3 - 2((1 - cos a)/2)
= 3 - (1 - cos a)
= 2 + cos a
Therefore, 1 + 2cos a can be expressed as the product of 2 and (cos a/2 + 1).
1 + 2cos a = 2(cos a/2 + 1)