The equation sin70° × sin20° sin40° × sin30° is a fascinating trigonometric identity that can be explored through various trigonometric identities and properties. Let's break down the solution step by step, using degrees for clarity.
Understanding the Identity
First, we need to clarify the identity and ensure it is correct within the context of degrees:
sin70° × sin20° sin40° × sin30°
By revisiting the trigonometric properties, specifically using the product-to-sum identities, we can prove this identity.
Using Trigonometric Relations
Method 1: Using the Cosine Difference Formula
According to the cosine difference formula, we have:
cos(A - B) cosAcosB sinAsinB
Setting (A 70^circ) and (B 20^circ), we get:
cos(70^circ - 20^circ) cos70^circ cos20^circ sin70^circ sin20^circ
Since (A - B 50^circ), we have:
cos50^circ cos70^circ cos20^circ sin70^circ sin20^circ
Using the co-function identity (cos50^circ sin40^circ), we get:
sin40^circ sin70^circ cos20^circ sin70^circ sin20^circ
Factoring out (sin70^circ) on the right-hand side, we get:
sin40^circ sin70^circ (cos20^circ sin20^circ)
Using the co-function identity again, (sin70^circ cos20^circ), we get:
sin40^circ cos20^circ (cos20^circ sin20^circ)
Simplifying, we get:
sin40^circ cos20^circ cos20^circ cos20^circ sin20^circ
This is equivalent to:
sin40^circ 2 cos20^circ cos20^circ 2 cos20^circ sin20^circ / 2
Which simplifies to:
sin40^circ 2 cos20^circ cos20^circ / 2 2 cos20^circ sin20^circ / 2
This is:
sin40^circ cos20^circ cos20^circ cos20^circ sin20^circ
Which is:
sin40^circ cos20^circ (cos20^circ sin20^circ)
Finally, we can use the identity (sin30^circ 1/2) to simplify:
sin30^circ sin40^circ 1/2 sin40^circ 1/2 (cos20^circ (cos20^circ sin20^circ))
This confirms that:
sin70^circ sin20^circ sin40^circ sin30^circ
Conclusion
Thus, the trigonometric identity sin70° × sin20° sin40° × sin30° holds true, as proven using trigonometric identities and properties of co-functions. The solution shows that when working in degrees, the identity is valid, but care must be taken when using radians, as the relationship changes.