simplify
cos(30+x)+cos(30-x)
Why did the angle go to a comedy club? It wanted to find its cosine! But in all seriousness, let's simplify that expression. The cosine function is an even function, which means that cos(-x) is equal to cos(x). So, we can rewrite the expression as cos(30+x) + cos(30-(-x)), which simplifies to cos(30+x) + cos(30+x). Combining the two terms, we'll get 2cos(30+x). And that's as simplified as it gets!
To simplify the expression cos(30+x) + cos(30-x), we can use the trigonometric identity:
cos(A + B) + cos(A - B) = 2 cos(A) cos(B)
Let's substitute A = 30 and B = x:
cos(30 + x) + cos(30 - x) = 2 cos(30) cos(x)
The cosine of 30 degrees is equal to √3/2. Therefore:
2 cos(30) cos(x) = 2 (√3/2) cos(x) = √3 cos(x)
So, the simplified expression is √3 cos(x).
To simplify the expression cos(30+x) + cos(30-x), we can make use of the trigonometric identity known as the sum of angles formula, which states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
First, let's apply this formula to the expression cos(30+x):
cos(30+x) = cos(30)cos(x) - sin(30)sin(x)
Next, let's apply the sum of angles formula to the expression cos(30-x):
cos(30-x) = cos(30)cos(-x) - sin(30)sin(-x)
Keep in mind that cos(x) = cos(-x) and sin(-x) = -sin(x).
Now, let's substitute these results back into the original expression:
cos(30+x) + cos(30-x) = (cos(30)cos(x) - sin(30)sin(x)) + (cos(30)cos(-x) - sin(30)sin(-x))
Using the commutative property of addition, we can group like terms together:
cos(30+x) + cos(30-x) = cos(30)cos(x) + cos(30)cos(-x) - sin(30)sin(x) - sin(30)sin(-x)
Now, recall that cos(-x) = cos(x) and sin(-x) = -sin(x):
cos(30+x) + cos(30-x) = cos(30)cos(x) + cos(30)cos(x) - sin(30)sin(x) + sin(30)sin(x)
Combining like terms, we have:
cos(30+x) + cos(30-x) = 2cos(30)cos(x)
Finally, since cos(30) = √3/2, we can substitute this value into the equation:
cos(30+x) + cos(30-x) = 2 * (√3/2) * cos(x)
Simplifying further, we get:
cos(30+x) + cos(30-x) = √3cos(x)
Therefore, the simplified expression is √3cos(x).
Use the identities:
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
and after that use the fact that cos(30°)=(√3)/2