Simplifying Trigonometric Expressions

This question asks us to simplify a given trigonometric expression involving powers of sine and cosine. To simplify, we will use fundamental trigonometric identities and algebraic manipulation.

Understanding the Expression

The expression we need to simplify is:

\(\rm \frac{\sin^4 \theta+\cos ^4\theta}{1-2\sin^2\theta.\cos^2\theta}\)

Let's look at the numerator and denominator separately.

Simplifying the Numerator: \(\sin^4 \theta+\cos ^4\theta\)

We can rewrite the numerator \(\sin^4 \theta+\cos ^4\theta\) using algebraic identities. Recall the identity \((a+b)^2 = a^2 + b^2 + 2ab\). From this, we can write \(a^2 + b^2 = (a+b)^2 - 2ab\).

Let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\).

Then, \(\sin^4 \theta+\cos ^4\theta = (\sin^2 \theta)^2 + (\cos^2 \theta)^2\).

Applying the identity \(a^2 + b^2 = (a+b)^2 - 2ab\), we get:

\(\sin^4 \theta+\cos ^4\theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2(\sin^2 \theta)(\cos^2 \theta)\)

Now, we use the fundamental trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

Substitute this into the expression:

\(\sin^4 \theta+\cos ^4\theta = (1)^2 - 2\sin^2 \theta \cos^2 \theta\)

\(\sin^4 \theta+\cos ^4\theta = 1 - 2\sin^2 \theta \cos^2 \theta\)

So, the simplified form of the numerator is \(1 - 2\sin^2 \theta \cos^2 \theta\).

Analyzing the Denominator: \(1-2\sin^2\theta.\cos^2\theta\)

The denominator of the original expression is \(1-2\sin^2\theta.\cos^2\theta\). This is exactly the same as the simplified form of the numerator we found.

Putting it Together

Now substitute the simplified numerator back into the original expression:

\(\rm \frac{\sin^4 \theta+\cos ^4\theta}{1-2\sin^2\theta.\cos^2\theta} = \frac{1 - 2\sin^2 \theta \cos^2 \theta}{1 - 2\sin^2 \theta \cos^2 \theta}\)

Assuming the denominator is not zero (which is generally true for real \(\theta\) as explained below), the expression simplifies to:

\(\frac{1 - 2\sin^2 \theta \cos^2 \theta}{1 - 2\sin^2 \theta \cos^2 \theta} = 1\)

Condition for Denominator Not Being Zero

The denominator \(1-2\sin^2\theta.\cos^2\theta\) would be zero if \(1 = 2\sin^2\theta.\cos^2\theta\). We know that \(\sin(2\theta) = 2\sin\theta\cos\theta\), so \(\sin^2(2\theta) = 4\sin^2\theta\cos^2\theta\), which means \(\sin^2\theta\cos^2\theta = \frac{\sin^2(2\theta)}{4}\). Substituting this, we get \(1 = 2 \left(\frac{\sin^2(2\theta)}{4}\right) = \frac{\sin^2(2\theta)}{2}\). This means \(\sin^2(2\theta) = 2\). However, the maximum value of \(\sin^2\) of any angle is 1. Since \(\sin^2(2\theta)\) cannot be greater than 1, \(\sin^2(2\theta) = 2\) has no real solution for \(\theta\). Therefore, the denominator \(1-2\sin^2\theta.\cos^2\theta\) is never zero for real values of \(\theta\).

Conclusion

The simplified value of the expression \(\rm \frac{\sin^4 \theta+\cos ^4\theta}{1-2\sin^2\theta.\cos^2\theta}\) is 1.

Reviewing the Options

• 1. 1
• 2. 2
• 3. -1
• 4. 0

Our simplified value is 1, which matches option 1.

Revision Table: Trigonometric Identities

Additional Information: Simplifying Complex Trigonometric Expressions

Simplifying trigonometric expressions often involves recognizing patterns and applying fundamental identities. Here are some tips:

• Look for opportunities to use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).
• Try to factor expressions or expand squares using algebraic identities.
• Convert all terms to sine and cosine if possible, especially in expressions involving tan, cot, sec, or cosec.
• Look for double angle or other compound angle identities if angles like \(2\theta\) or \(\theta/2\) appear.
• If the expression involves powers like \(\sin^4 \theta\) or \(\cos^4 \theta\), think about how they relate to \(\sin^2 \theta\) and \(\cos^2 \theta\). The algebraic identity \(a^2+b^2=(a+b)^2-2ab\) is very useful here.
• Always check if the denominator can be zero, although in many standard problems, it will be designed to be non-zero for real values of the variable.
• Practice is key! Work through various examples to become comfortable with different techniques.

By applying these strategies, we can simplify complex trigonometric expressions effectively.