Understanding the Trigonometric Expression

The problem asks us to simplify a given trigonometric expression and find which of the options it is equal to. The expression involves several trigonometric ratios like tan, sec, cot, and cosec. To simplify complex trigonometric expressions, it is often helpful to convert all terms into their basic sine and cosine forms.

Step-by-Step Simplification of the Expression

The given expression is:

\(\frac{(1 + tan\theta + sec\theta)(1 + cot\theta - cosec\theta)}{(sec\theta + tan\theta)(1 - sin\theta)}\)

Convert to Sine and Cosine

Let's convert each trigonometric ratio into its equivalent sine and cosine form:

• \(tan\theta = \frac{sin\theta}{cos\theta}\)
• \(sec\theta = \frac{1}{cos\theta}\)
• \(cot\theta = \frac{cos\theta}{sin\theta}\)
• \(cosec\theta = \frac{1}{sin\theta}\)

Simplify the Numerator

The numerator is \((1 + tan\theta + sec\theta)(1 + cot\theta - cosec\theta)\). Substitute the sine and cosine forms:

\((1 + \frac{sin\theta}{cos\theta} + \frac{1}{cos\theta})(1 + \frac{cos\theta}{sin\theta} - \frac{1}{sin\theta})\)

Combine terms within each parenthesis by finding a common denominator:

\((\frac{cos\theta + sin\theta + 1}{cos\theta})(\frac{sin\theta + cos\theta - 1}{sin\theta})\)

Now, multiply the two fractions. The numerator becomes \((cos\theta + sin\theta + 1)(sin\theta + cos\theta - 1)\). This looks like \((a+b)(a-b)\) where \(a = cos\theta + sin\theta\) and \(b = 1\). Using the difference of squares formula, \(a^2 - b^2\):

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

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

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

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

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

So, the numerator of the combined fraction is \((1 + 2sin\theta cos\theta) - 1 = 2sin\theta cos\theta\). The denominator of the combined fraction is \(cos\theta sin\theta\). Thus, the simplified numerator of the original expression is:

\(\frac{2sin\theta cos\theta}{cos\theta sin\theta} = 2\)

Simplify the Denominator

The denominator of the original expression is \((sec\theta + tan\theta)(1 - sin\theta)\). Substitute the sine and cosine forms for the first part:

\((\frac{1}{cos\theta} + \frac{sin\theta}{cos\theta})(1 - sin\theta)\)

Combine terms within the parenthesis:

\((\frac{1 + sin\theta}{cos\theta})(1 - sin\theta)\)

Multiply the fractions:

\(\frac{(1 + sin\theta)(1 - sin\theta)}{cos\theta}\)

The numerator is \((1 + sin\theta)(1 - sin\theta)\), which is in the form \((a+b)(a-b)\). Using the difference of squares formula, \(a^2 - b^2\):

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

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

\(cos^2\theta\)

So, the simplified denominator is:

\(\frac{cos^2\theta}{cos\theta} = cos\theta\)

Combine Simplified Numerator and Denominator

Now substitute the simplified numerator (which is 2) and the simplified denominator (which is \(cos\theta\)) back into the original expression:

\(\frac{2}{cos\theta}\)

Since \(sec\theta = \frac{1}{cos\theta}\), we can write this as:

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

Final Result

The simplified expression is \(2sec\theta\).

Revision Table: Key Trigonometric Identities

Additional Information on Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using fundamental identities to rewrite the expression in a simpler form. A common strategy, as used in this problem, is to convert all terms into sine and cosine. Other useful techniques include:

• Factoring expressions.
• Finding common denominators for fractions.
• Using Pythagorean identities \(sin^2\theta + cos^2\theta = 1\), \(1 + tan^2\theta = sec^2\theta\), \(1 + cot^2\theta = cosec^2\theta\).
• Using sum/difference, double angle, or half angle formulas if applicable.
• Multiplying by a conjugate.

Practice is key to becoming proficient in recognizing which identities and techniques to apply to simplify a given trigonometric expression.