Understanding the Trigonometric Expressions

We are given two initial expressions involving the angle \(\theta\):

• \(p = \text{cosec}\theta - \sin\theta\)
• \(q = \sec\theta - \cos\theta\)

The task is to determine the value of the expression \((p\sin\theta + q\cos\theta)\).

Simplifying the Expression for p

First, let's simplify the expression for \(p\). We know that \(\text{cosec}\theta\) is the reciprocal of \(\sin\theta\), meaning \(\text{cosec}\theta = \frac{1}{\sin\theta}\). Substituting this into the expression for \(p\):

\(p = \frac{1}{\sin\theta} - \sin\theta\)

To combine these terms, we use a common denominator, \(\sin\theta\):

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

Recall the fundamental Pythagorean trigonometric identity: \(\sin^2\theta + \cos^2\theta = 1\). Rearranging this gives \(1 - \sin^2\theta = \cos^2\theta\). Substituting this identity:

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

Simplifying the Expression for q

Next, we simplify the expression for \(q\). We know that \(\sec\theta\) is the reciprocal of \(\cos\theta\), meaning \(\sec\theta = \frac{1}{\cos\theta}\). Substituting this into the expression for \(q\):

\(q = \frac{1}{\cos\theta} - \cos\theta\)

Using a common denominator, \(\cos\theta\):

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

Again, using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), we can rearrange it to \(1 - \cos^2\theta = \sin^2\theta\). Substituting this identity:

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

Evaluating the Target Expression \((p\sin\theta + q\cos\theta)\)

Now we substitute the simplified forms of \(p\) and \(q\) into the expression we need to evaluate:

\((p\sin\theta + q\cos\theta) = \left(\frac{\cos^2\theta}{\sin\theta}\right) \sin\theta + \left(\frac{\sin^2\theta}{\cos\theta}\right) \cos\theta\)

We can see that \(\sin\theta\) cancels in the first term and \(\cos\theta\) cancels in the second term:

\((p\sin\theta + q\cos\theta) = \cos^2\theta + \sin^2\theta\)

Finally, we apply the fundamental Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) one last time:

\((p\sin\theta + q\cos\theta) = 1\)

Thus, the value of the expression \((p\sin\theta + q\cos\theta)\) is 1.