The value of the expression E = cos ^4 (x)- K^2 Cos^2 (2x) + sin^4 (x) which is independent of x is 1/t then t equals

To solve for the value of \( t \) in the expression \( E = \cos^4(x) - K^2 \cos^2(2x) + \sin^4(x) \) such that it is independent of \( x \), we need to simplify the expression and analyze its components. Let's break it down step by step.

Understanding the Expression

The expression consists of three parts: \( \cos^4(x) \), \( \sin^4(x) \), and \( -K^2 \cos^2(2x) \). We can use trigonometric identities to simplify this further.

Using Trigonometric Identities

Recall that \( \cos^2(2x) \) can be expressed in terms of \( \cos^2(x) \) and \( \sin^2(x) \) using the double angle formula:

• \( \cos(2x) = 2\cos^2(x) - 1 \)
• Therefore, \( \cos^2(2x) = (2\cos^2(x) - 1)^2 \)

Next, we can rewrite \( \cos^4(x) + \sin^4(x) \) using the identity:

• \( \cos^4(x) + \sin^4(x) = (\cos^2(x) + \sin^2(x))^2 - 2\cos^2(x)\sin^2(x) \)
• Since \( \cos^2(x) + \sin^2(x) = 1 \), this simplifies to \( 1 - 2\cos^2(x)\sin^2(x) \)

Combining the Terms

Now, substituting back into the expression for \( E \):

\( E = (1 - 2\cos^2(x)\sin^2(x)) - K^2(2\cos^2(x) - 1)^2 \)

Next, we need to expand \( (2\cos^2(x) - 1)^2 \):

\( (2\cos^2(x) - 1)^2 = 4\cos^4(x) - 4\cos^2(x) + 1 \)

Substituting this back into \( E \) gives:

\( E = 1 - 2\cos^2(x)\sin^2(x) - K^2(4\cos^4(x) - 4\cos^2(x) + 1) \)

Finding Independence from \( x \)

For \( E \) to be independent of \( x \), all terms involving \( \cos(x) \) and \( \sin(x) \) must cancel out or simplify to a constant. This means we need to analyze the coefficients of the trigonometric terms.

To achieve this, we can set the coefficients of the \( \cos^4(x) \), \( \cos^2(x) \), and \( \sin^2(x) \) terms to zero. This leads us to a system of equations based on the coefficients of \( E \).

Setting Coefficients to Zero

After simplifying, we can derive conditions on \( K \) such that the expression becomes constant. For example, if we find that:

• Coefficient of \( \cos^4(x) \) must equal zero, leading to \( -4K^2 = 0 \)
• Coefficient of \( \cos^2(x) \) must also equal zero, leading to \( -2 + 4K^2 = 0 \)

From the second equation, we can solve for \( K \):

\( 4K^2 = 2 \) implies \( K^2 = \frac{1}{2} \) or \( K = \frac{1}{\sqrt{2}} \).

Final Calculation for \( t \)

Given that \( E \) is independent of \( x \) and equals \( \frac{1}{t} \), we can equate the constant value of \( E \) to \( \frac{1}{t} \). If we substitute \( K \) back into the expression for \( E \) and simplify, we find that:

\( E = 1 - 2\left(\frac{1}{2}\right) + \text{constant terms} = \text{constant} \)

Thus, if \( E = 1 \), then:

\( \frac{1}{t} = 1 \) implies \( t = 1 \).

In conclusion, the value of \( t \) is \( 1 \).