Understanding the Value of Trigonometric Functions in a Given Equation

The problem presented is to find the value of a given trigonometric equation. The equation in question is:

Given Equation:

sin6x cos6x a2

Step-by-Step Solution:

Letrsquo;s break down this problem into manageable steps, using basic trigonometric identities and algebraic manipulations. We can start by expressing the given equation in simpler forms.

Step 1: Express the Equation in Terms of Squared Sine and Cosine

First, we rewrite the equation:

sin6x cos6x sin2x cos2x sin4x cos4x

Step 2: Further Simplification Using Trigonometric Identities

We can simplify it further by using the double-angle identity for sine, sin2x (1 - cos2x)/2.

Letrsquo;s denote the angle 2x as y for simplicity. Then, the equation becomes:

sin6x cos6x (1 - cos2y)2 cos4y / 16

Since cos2y can be expressed using another identity, we have:

cos2y (1 cos2y)/2

Therefore, we can substitute and simplify further:

sin6x cos6x (1 - (1 cos2y)/2)2 ((1 cos2y)/2)2 / 16

Step 3: Simplifying the Expression

Simplifying the above expression, we get:

sin6x cos6x (1 - 1/2 - cos2y/2)2 (1/4 cos2y/4 cos22y/4) / 16

This can be further simplified as:

sin6x cos6x (1/2 - cos2y/2)2 (1/16 cos2y/16 cos22y/16)

Step 4: Identifying the Value of a

Given that sin6x cos6x a2, we have:

a2 (1/2 - cos2y/2)2 (1/16 cos2y/16 cos22y/16)

We know from trigonometric identities that cos22y can have a maximum value of 1 and a minimum value of 0. Therefore, the expression simplifies to:

a2 (1/2 - 1/2)2 (1/16 1/16 1/16) (0)2 * (3/16) 0 * (3/16) 0

However, the given solution suggests a more complex expression, leading us to:

sin22x 4(1 - a2)

Given that sin22x 1 - (1 - cos22x) and knowing that sin22x 1 - cos22x, we have:

1 - cos22x 4(1 - a2)

Since cos22x can be 0 to 1, we can solve for a2 as follows:

1 - 0 4(1 - a2)

a2 1/4

Therefore, a can be:

a sqrt(3/4) sqrt(3)/2, a -sqrt(3)/2 and a 1

Conclusion:

The possible values of a are sqrt(3)/2, -sqrt(3)/2, and 1. This result is obtained from the given trigonometric identities and the algebraic manipulation of the equation.

Keywords:

trigonometric identities, trigonometric functions, algebraic manipulation