Simplifying Trigonometric Expressions: A Step-by-Step Guide

Let's simplify the given trigonometric expression:

\[\frac{1+\sin^4 \theta+\cos^4\theta}{\cos^2 \theta+\sin^4\theta}\]

We will simplify the numerator and the denominator separately using fundamental trigonometric identities.

Simplifying the Numerator: \(1+\sin^4 \theta+\cos^4\theta\)

Recall the basic identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

We can rewrite \(\sin^4 \theta + \cos^4 \theta\) using this identity:

\[\sin^4 \theta + \cos^4 \theta = (\sin^2 \theta)^2 + (\cos^2 \theta)^2\]

This looks like the form \(a^2 + b^2\). We know \(a^2 + b^2 = (a+b)^2 - 2ab\).

So, let \(a = \sin^2 \theta\) and \(b = \cos^2 \theta\):

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

Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

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

Now substitute this back into the numerator:

Numerator = \(1 + (\sin^4 \theta + \cos^4 \theta)\)

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

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

Numerator = \(2 - 2 \sin^2 \theta \cos^2 \theta\)

Numerator = \(2(1 - \sin^2 \theta \cos^2 \theta)\)

Analyzing the Denominator: \(\cos^2 \theta+\sin^4\theta\)

The denominator is \(\cos^2 \theta+\sin^4\theta\).

Let's examine the simplified numerator: \(2(1 - \sin^2 \theta \cos^2 \theta)\). Can we relate the expression \(1 - \sin^2 \theta \cos^2 \theta\) to the denominator?

Recall \(1 = \sin^2 \theta + \cos^2 \theta\).

So, \(1 - \sin^2 \theta \cos^2 \theta = (\sin^2 \theta + \cos^2 \theta) - \sin^2 \theta \cos^2 \theta\). This doesn't immediately look like the denominator.

Let's try to show that \(1 - \sin^2 \theta \cos^2 \theta\) can be written in a form related to the denominator \(cos^2 \theta + sin^4 \theta\). Consider the identity we need to establish:

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

Let's work from the right side and try to reach the left side using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\):

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

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

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

This still doesn't look like \(1 - \sin^2 \theta \cos^2 \theta\).

Let's re-examine the relationship derived from the correct answer option being 2. If the expression simplifies to 2, it means:

\[\frac{1+\sin^4 \theta+\cos^4\theta}{\cos^2 \theta+\sin^4\theta} = 2\]

This implies that the numerator is equal to 2 times the denominator:

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

We know the numerator simplifies to \(2 - 2 \sin^2 \theta \cos^2 \theta\). So we need to check if:

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

Divide both sides by 2:

\[1 - \sin^2 \theta \cos^2 \theta = \cos^2 \theta+\sin^4\theta\]

We need to verify if this identity is true for all \(\theta\) where the expression is defined.

Let's rearrange the terms to check:

\[1 - \cos^2 \theta = \sin^2 \theta \cos^2 \theta + \sin^4 \theta\]

We know \(1 - \cos^2 \theta = \sin^2 \theta\). So we need to check if:

\[\sin^2 \theta = \sin^2 \theta \cos^2 \theta + \sin^4 \theta\]

Factor out \(\sin^2 \theta\) from the right side:

\[\sin^2 \theta = \sin^2 \theta (\cos^2 \theta + \sin^2 \theta)\]

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\):

\[\sin^2 \theta = \sin^2 \theta (1)\]

\[\sin^2 \theta = \sin^2 \theta\]

This is true for all \(\theta\)! Therefore, the identity \(1 - \sin^2 \theta \cos^2 \theta = \cos^2 \theta+\sin^4\theta\) is correct.

Final Simplification

We found that:

• Numerator = \(1+\sin^4 \theta+\cos^4\theta = 2(1 - \sin^2 \theta \cos^2 \theta)\)
• Denominator = \(\cos^2 \theta+\sin^4\theta\)
• And we verified that \(1 - \sin^2 \theta \cos^2 \theta = \cos^2 \theta+\sin^4\theta\)

Substitute the equivalent expression for \(1 - \sin^2 \theta \cos^2 \theta\) into the numerator:

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

Now, substitute this back into the original expression:

\[\frac{2(\cos^2 \theta+\sin^4\theta)}{\cos^2 \theta+\sin^4\theta}\]

Assuming the denominator \(\cos^2 \theta+\sin^4\theta\) is not equal to zero, we can cancel the common term:

\[\frac{2\cancel{(\cos^2 \theta+\sin^4\theta)}}{\cancel{(\cos^2 \theta+\sin^4\theta)}} = 2\]

Thus, the simplified expression is 2.

Summary of Steps

To simplify the expression, we followed these steps:

1. Simplified the numerator \(1+\sin^4 \theta+\cos^4\theta\) using the identity \(\sin^2 \theta+\cos^2\theta=1\) to get \(2(1 - \sin^2 \theta \cos^2 \theta)\).
2. Examined the denominator \(\cos^2 \theta+\sin^4\theta\).
3. Proved the identity \(1 - \sin^2 \theta \cos^2 \theta = \cos^2 \theta+\sin^4\theta\) by substituting \(\cos^2 \theta = 1 - \sin^2 \theta\) and simplifying to \(\sin^2 \theta = \sin^2 \theta\).
4. Substituted the equivalent expression into the numerator, resulting in \(2(\cos^2 \theta+\sin^4\theta)\).
5. Divided the numerator by the denominator, cancelling the common term to get 2.

The final answer is 2.

Additional Information: Trigonometric Identities for Simplification

Simplifying trigonometric expressions often relies on recognizing and applying fundamental identities. Here are some important ones:

• Pythagorean Identities:
  • \(\sin^2 \theta + \cos^2 \theta = 1\)
  • \(1 + \tan^2 \theta = \sec^2 \theta\)
  • \(1 + \cot^2 \theta = \csc^2 \theta\)
• Reciprocal Identities:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
• Quotient Identities:
  • \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
  • \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Manipulating expressions involving higher powers like \(\sin^4 \theta\) or \(\cos^4 \theta\) often involves using the square identities repeatedly or techniques like the one used in this problem where \(a^4+b^4 = (a^2+b^2)^2 - 2a^2b^2\). Recognizing patterns and equivalent forms of expressions is key to successful simplification.