Evaluating Trigonometric Expression

The question asks us to find the value of the expression \(\frac{1+\tan \theta}{1-\cot \theta}\) given that \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\).

To evaluate this expression, we first need to find the values of \(\tan \theta\) and \(\cot \theta\) using the given values of \(\sin \theta\) and \(\cos \theta\).

Finding tan θ and cot θ using Sine and Cosine

The trigonometric ratios \(\tan \theta\) and \(\cot \theta\) can be expressed in terms of \(\sin \theta\) and \(\cos \theta\) using the following identities:

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

Let's calculate \(\tan \theta\):

\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}}\]

To simplify this fraction, we can multiply the numerator by the reciprocal of the denominator:

\[\tan \theta = \frac{3}{5} \times \frac{5}{4} = \frac{3 \times 5}{5 \times 4} = \frac{15}{20}\]

Simplifying the fraction \(\frac{15}{20}\) by dividing both numerator and denominator by 5:

\[\tan \theta = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}\]

Now, let's calculate \(\cot \theta\):

\[\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{4}{5}}{\frac{3}{5}}\]

Multiplying the numerator by the reciprocal of the denominator:

\[\cot \theta = \frac{4}{5} \times \frac{5}{3} = \frac{4 \times 5}{5 \times 3} = \frac{20}{15}\]

Simplifying the fraction \(\frac{20}{15}\) by dividing both numerator and denominator by 5:

\[\cot \theta = \frac{20 \div 5}{15 \div 5} = \frac{4}{3}\]

Alternatively, using \(\cot \theta = \frac{1}{\tan \theta}\):

\[\cot \theta = \frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3} = \frac{4}{3}\]

So, we have \(\tan \theta = \frac{3}{4}\) and \(\cot \theta = \frac{4}{3}\).

Substituting Values and Simplifying the Expression

Now we substitute these values into the given expression \(\frac{1+\tan \theta}{1-\cot \theta}\):

\[\frac{1+\tan \theta}{1-\cot \theta} = \frac{1+\frac{3}{4}}{1-\frac{4}{3}}\]

Let's simplify the numerator and the denominator separately.

Numerator: \(1+\frac{3}{4}\)

To add 1 and \(\frac{3}{4}\), we express 1 as a fraction with a denominator of 4:

\[1+\frac{3}{4} = \frac{4}{4}+\frac{3}{4} = \frac{4+3}{4} = \frac{7}{4}\]

Denominator: \(1-\frac{4}{3}\)

To subtract \(\frac{4}{3}\) from 1, we express 1 as a fraction with a denominator of 3:

\[1-\frac{4}{3} = \frac{3}{3}-\frac{4}{3} = \frac{3-4}{3} = \frac{-1}{3}\]

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

\[\frac{\frac{7}{4}}{\frac{-1}{3}}\]

To divide fractions, we multiply the numerator fraction by the reciprocal of the denominator fraction:

\[\frac{7}{4} \div \frac{-1}{3} = \frac{7}{4} \times \frac{3}{-1}\]

Multiply the numerators together and the denominators together:

\[\frac{7 \times 3}{4 \times (-1)} = \frac{21}{-4}\]

The value can be written as \(\frac{-21}{4}\).

Final Answer

The value of the expression \(\frac{1+\tan \theta}{1-\cot \theta}\) is \(\frac{-21}{4}\).

Revision Table: Key Trigonometric Ratios

Here's a quick look at the basic trigonometric ratios derived from a right-angled triangle with an angle \(\theta\):

Additional Information: Understanding Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where the expressions are defined. They are fundamental in simplifying trigonometric expressions and solving trigonometric equations.

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

These identities are powerful tools for manipulating trigonometric expressions and are frequently used in solving problems like the one discussed.