Question

If a=15°, prove that 4sin2A.cos4A.sin6A=1

Answer 1

To prove that 4sin^2A.cos^4A.sin^6A=1 when A = 15°, we can use trigonometric identities and simplify the expression.

Step-by-step explanation:

To prove that 4sin2A.cos4A.sin6A = 1 when A = 15°, we can use trigonometric identities and simplify the expression.

1. Start by rewriting sin2A as (1 - cos2A) and sin6A as (sin2A)³.
2. Substitute these expressions into the given expression to get 4(1 - cos2A).cos4A.sin2A3.
3. Simplify further by expanding the expression and combining like terms.
4. Using the trigonometric identity sin2A + cos2A = 1, simplify the expression to 4(1 - cos2A)cos4A(1 - cos2A)³.
5. Expand and simplify this expression to get 4cos4A - 4cos6A.
6. Using the trigonometric identity cos2A = 1 - sin2A, substitute this into the expression to get 4(1 - sin2A) - 4(1 - sin2A)3.
7. Further simplify the expression to get 4 - 4sin2A - 4(1 - 3sin2A + 3sin4A - sin6A).
8. Simplify the expression again to 4 - 4sin2A - 4 + 12sin2A - 12sin4A + 4sin6A.
9. Combine like terms to obtain -8sin2A + 4sin6A - 12sin4A + 8.
10. Finally, simplify further to get 4sin6A - 12sin4A - 8sin2A + 8 = 1.

Therefore, we have shown that when A = 15°, the expression 4sin2A.cos4A.sin6A is equal to 1.