Final answer:
Using trigonometric identities, sin 70° + sin 50° simplifies to √3 cos(10°), which is not equal to 3 cos 10°. Therefore, the original equation is not true, and none of the provided options are correct.
Step-by-step explanation:
To solve the given problem, we will use the sum-to-product trigonometric identities. Specifically, the identity for the sum of two sine functions:
sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2).
Applying this identity to sin 70° + sin 50°:
sin 70° + sin 50° = 2 sin((70° + 50°)/2) cos((70° - 50°)/2)
= 2 sin(60°) cos(10°)
Now, using the fact that sin(60°) = √3/2, the equation becomes:
= 2 (√3/2) cos(10°)
= √3 cos(10°)
The original problem was:
sin 70° + sin 50° = 3 cos 10°
If we replace the left side of the equation with what we found, we get:
√3 cos(10°) = 3 cos 10°
To solve for √3, we divide both sides by cos(10°):
√3 = 3
Since we know that √3 is not equal to 3, we can conclude that the original statement is not true with any of the provided options (A) 100, (B) 200, (C) 300, or (D) 400.