Prove that : sin 10 degree × sin 30 degree × sin 50 degree × sin 70 de...
**Proof:**
To prove the given equation, we will use the trigonometric identity:
*sin(A) × sin(B) = (1/2) [cos(A - B) - cos(A + B)]*
Let's start by rewriting the given equation using this identity:
*sin(10°) × sin(30°) × sin(50°) × sin(70°) = (1/2) [cos(30° - 10°) - cos(30° + 10°)] × (1/2) [cos(70° - 50°) - cos(70° + 50°)]*
Now, let's simplify the equation step by step:
**Step 1: Simplifying the first bracket**
cos(30° - 10°) - cos(30° + 10°) = cos(20°) - cos(40°)
Using the trigonometric identity:
cos(A - B) - cos(A + B) = -2 sin(A) sin(B)
cos(20°) - cos(40°) = -2 sin(20°) sin(40°)
**Step 2: Simplifying the second bracket**
cos(70° - 50°) - cos(70° + 50°) = cos(20°) - cos(120°)
Using the trigonometric identity:
cos(A - B) - cos(A + B) = -2 sin(A) sin(B)
cos(20°) - cos(120°) = -2 sin(20°) sin(120°)
**Step 3: Combining the simplified brackets**
(-2 sin(20°) sin(40°)) × (-2 sin(20°) sin(120°))
= 4 sin(20°) sin(40°) sin(120°)
**Step 4: Using the trigonometric identity**
sin(120°) = sqrt(3)/2
Therefore, the equation becomes:
4 sin(20°) sin(40°) (sqrt(3)/2)
**Step 5: Simplifying further**
sin(20°) = sin(180° - 20°) = sin(160°)
Using the trigonometric identity:
sin(A) = sin(180° - A)
sin(160°) = sin(20°)
Therefore, the equation becomes:
4 sin(20°) sin(40°) (sqrt(3)/2)
= 4 sin(40°) sin(20°) (sqrt(3)/2)
**Step 6: Applying the trigonometric identity again**
sin(40°) = sin(180° - 40°) = sin(140°)
Using the trigonometric identity:
sin(A) = sin(180° - A)
sin(140°) = sin(40°)
Therefore, the equation becomes:
4 sin(40°) sin(20°) (sqrt(3)/2)
= 4 sin(20°) sin(40°) (sqrt(3)/2)
**Step 7: Final Simplification**
We can see that the equation has the same terms as its original form:
4 sin(20°) sin(40°) (sqrt(3)/2)
Therefore, the final equation is:
sin(10°) × sin(30
Prove that : sin 10 degree × sin 30 degree × sin 50 degree × sin 70 de...
Use firmula 2 sin A sin B and 2 CosA sin B
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