Prove that: [tex]$\sin ^8 A+\cos ^8 A=1-\frac{2}{\sin ^2} 2 A+\frac{1}{8} \sin ^4 2 A$[/tex]
Answer (2)
To prove the identity sin 8 A + cos 8 A = 1 − sin 2 2 A + 8 1 sin 4 2 A , we rewrite the left-hand side and simplify using trigonometric identities. After substituting and expanding, we find both sides are equal, completing the proof. This mathematical identity illustrates the interplay between powers of sine and cosine functions and their double angle identities.
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Rewrite sin 8 A + cos 8 A as ( sin 4 A + cos 4 A ) 2 − 2 sin 4 A cos 4 A .
Simplify sin 4 A + cos 4 A to 1 − 2 sin 2 A cos 2 A .
Substitute and expand to get 1 − 4 sin 2 A cos 2 A + 2 sin 4 A cos 4 A .
Apply the double angle identity to obtain 1 − sin 2 2 A + 8 1 sin 4 2 A , proving the identity: sin 8 A + cos 8 A = 1 − sin 2 2 A + 8 1 sin 4 2 A .
Explanation
Understanding the Problem We want to prove the trigonometric identity: sin 8 A + cos 8 A = 1 − sin 2 2 A + 8 1 sin 4 2 A .
Rewriting the LHS Let's start with the left-hand side (LHS) of the equation: sin 8 A + cos 8 A . We can rewrite this as ( sin 4 A + cos 4 A ) 2 − 2 sin 4 A cos 4 A .
Simplifying Now, let's express sin 4 A + cos 4 A as ( sin 2 A + cos 2 A ) 2 − 2 sin 2 A cos 2 A . Since sin 2 A + cos 2 A = 1 , this simplifies to 1 − 2 sin 2 A cos 2 A .
Substitution Substitute this back into the expression from step 2: ( 1 − 2 sin 2 A cos 2 A ) 2 − 2 sin 4 A cos 4 A .
Expanding Expand the expression: 1 − 4 sin 2 A cos 2 A + 4 sin 4 A cos 4 A − 2 sin 4 A cos 4 A = 1 − 4 sin 2 A cos 2 A + 2 sin 4 A cos 4 A .
Using Double Angle Identity Now, let's use the double angle identity sin 2 A = 2 sin A cos A . Thus, sin 2 2 A = 4 sin 2 A cos 2 A , so sin 2 A cos 2 A = 4 1 sin 2 2 A .
Final Simplification Substitute sin 2 A cos 2 A = 4 1 sin 2 2 A into the expression: 1 − 4 ( 4 1 sin 2 2 A ) + 2 ( 4 1 sin 2 2 A ) 2 = 1 − sin 2 2 A + 2 ( 16 1 sin 4 2 A ) = 1 − sin 2 2 A + 8 1 sin 4 2 A .
Conclusion Comparing the simplified LHS with the right-hand side (RHS) of the equation, we have 1 − sin 2 2 A + 8 1 sin 4 2 A = 1 − sin 2 2 A + 8 1 sin 4 2 A . Since LHS = RHS, the identity is proven.
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and computer graphics. For instance, in physics, they are used to simplify complex equations describing wave phenomena, such as light and sound. In engineering, they are crucial for analyzing oscillating systems and signal processing. In computer graphics, trigonometric identities help in creating realistic animations and transformations of objects in 3D space. Understanding and manipulating these identities allows for efficient and accurate modeling of real-world phenomena.
