Evaluating the Trigonometric Expression

Let's evaluate the given trigonometric expression step-by-step by simplifying the numerator and the denominator separately.

The given expression is:

\(\frac{{{{\cos }^4}\theta + {{\sin }^4}\theta + 2{{\sin }^2}\theta {{\cos }^2}\theta }}{{cosec\theta \sec\theta \left( {\sin\theta + \cos\theta - 1} \right)\left( {\sin\theta + \cos\theta + 1} \right)}}\)

Simplifying the Numerator

The numerator is \({{\cos }^4}\theta + {{\sin }^4}\theta + 2{{\sin }^2}\theta {{\cos }^2}\theta\). This expression resembles the expansion of a perfect square formula, \((a+b)^2 = a^2 + b^2 + 2ab\).

Let \(a = \cos^2\theta\) and \(b = \sin^2\theta\). Then the numerator becomes:

\((\cos^2\theta)^2 + (\sin^2\theta)^2 + 2(\sin^2\theta)(\cos^2\theta)\)

This is exactly in the form \(a^2 + b^2 + 2ab\), so it simplifies to \((a+b)^2\):

\((\cos^2\theta + \sin^2\theta)^2\)

We know the fundamental trigonometric identity: \(\sin^2\theta + \cos^2\theta = 1\).

Substituting this identity into the simplified numerator:

\((1)^2 = 1\)

So, the numerator simplifies to 1.

Simplifying the Denominator

The denominator is \(cosec\theta \sec\theta \left( {\sin\theta + \cos\theta - 1} \right)\left( {\sin\theta + \cos\theta + 1} \right)\).

Let's first simplify the part with the parentheses: \(\left( {\sin\theta + \cos\theta - 1} \right)\left( {\sin\theta + \cos\theta + 1} \right)\). This looks like the difference of squares formula, \((A-B)(A+B) = A^2 - B^2\).

Let \(A = \sin\theta + \cos\theta\) and \(B = 1\). Then the expression becomes:

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

Expand the square of the sum: \((\sin\theta + \cos\theta)^2 = \sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta\).

So, the expression is:

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

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

\((1 + 2\sin\theta\cos\theta) - 1\)

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

\(= 2\sin\theta\cos\theta\)

Now, let's look at the \(cosec\theta \sec\theta\) part. We know that \(cosec\theta = \frac{1}{\sin\theta}\) and \(sec\theta = \frac{1}{\cos\theta}\).

So, \(cosec\theta \sec\theta = \frac{1}{\sin\theta} \cdot \frac{1}{\cos\theta} = \frac{1}{\sin\theta\cos\theta}\).

Now, combine all parts of the denominator:

\(Denominator = (cosec\theta \sec\theta) \times \left( (\sin\theta + \cos\theta)^2 - 1 \right)\)

\(Denominator = \frac{1}{\sin\theta\cos\theta} \times (2\sin\theta\cos\theta)\)

Assuming \(\sin\theta \neq 0\) and \(\cos\theta \neq 0\), we can cancel out the \(\sin\theta\cos\theta\) term:

\(Denominator = 2\)

So, the denominator simplifies to 2.

Calculating the Final Value

Now, we substitute the simplified numerator and denominator back into the original expression:

\(\frac{Numerator}{Denominator} = \frac{1}{2}\)

Thus, the value of the given trigonometric expression is \(\frac{1}{2}\).

Summary of Steps

• The numerator \({{\cos }^4}\theta + {{\sin }^4}\theta + 2{{\sin }^2}\theta {{\cos }^2}\theta\) was recognized as a perfect square, \((\cos^2\theta + \sin^2\theta)^2\).
• Using the identity \(\sin^2\theta + \cos^2\theta = 1\), the numerator simplified to \(1^2 = 1\).
• The part of the denominator \(\left( {\sin\theta + \cos\theta - 1} \right)\left( {\sin\theta + \cos\theta + 1} \right)\) was simplified using the difference of squares identity as \((\sin\theta + \cos\theta)^2 - 1\).
• Expanding \((\sin\theta + \cos\theta)^2\) gave \(\sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta\).
• Using \(\sin^2\theta + \cos^2\theta = 1\), this part became \(1 + 2\sin\theta\cos\theta - 1 = 2\sin\theta\cos\theta\).
• The terms \(cosec\theta \sec\theta\) were written as \(\frac{1}{\sin\theta\cos\theta}\).
• Multiplying these parts of the denominator \(\frac{1}{\sin\theta\cos\theta} \times 2\sin\theta\cos\theta\) resulted in 2.
• The final value of the expression is the ratio of the simplified numerator to the simplified denominator, which is \(\frac{1}{2}\).

Revision Table: Key Trigonometric Identities

Additional Information: Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using fundamental identities and algebraic manipulations. Here are some tips:

• Know Your Identities: Be familiar with Pythagorean identities (\(\sin^2\theta + \cos^2\theta = 1\)), reciprocal identities (\(cosec\theta = 1/\sin\theta\), \(sec\theta = 1/\cos\theta\)), quotient identities (\(tan\theta = \sin\theta/\cos\theta\)), and angle sum/difference formulas.
• Look for Algebraic Patterns: Expressions might fit algebraic formulas like \((a+b)^2\), \((a-b)^2\), \(a^2-b^2\), etc.
• Convert to Sine and Cosine: If stuck, converting all terms (like tan, cot, sec, cosec) into sine and cosine can simplify the expression.
• Combine Fractions: If there are multiple fractions, find a common denominator.
• Factorize: Look for common factors or factor using algebraic techniques.
• Practice: The more you practice simplifying, the better you become at recognizing patterns and choosing the right identities.

In this problem, recognizing the numerator as a perfect square and the denominator's parenthetical part as a difference of squares greatly simplified the process. Converting cosec and sec to their sine and cosine forms was also crucial.