To solve the given expression [tex]\((\operatorname{cosec} \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\)[/tex], let's proceed step-by-step using a specific angle [tex]\(\theta = \frac{\pi}{4}\)[/tex]:
### Step-by-Step Solution
1. Determine the standard trigonometric values for [tex]\(\theta = \frac{\pi}{4}\)[/tex]:
- [tex]\(\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex]
- [tex]\(\cot \left(\frac{\pi}{4}\right) = 1\)[/tex]
- [tex]\(\operatorname{cosec} \left(\frac{\pi}{4}\right) = \frac{1}{\sin \left(\frac{\pi}{4}\right)} = \sqrt{2}\)[/tex]
- [tex]\(\sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \sqrt{2}\)[/tex]
2. Substitute these values into the expression:
- [tex]\( \operatorname{cosec} \theta - \sin \theta = \sqrt{2} - \frac{\sqrt{2}}{2} = \sqrt{2} \left(1 - \frac{1}{2}\right) = \sqrt{2} \times \frac{1}{2} = \frac{\sqrt{2}}{2} \)[/tex]
- [tex]\(\sec \theta - \cos \theta = \sqrt{2} - \frac{\sqrt{2}}{2} = \sqrt{2} \left(1 - \frac{1}{2}\right) = \sqrt{2} \times \frac{1}{2} = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\tan \theta + \cot \theta = 1 + 1 = 2\)[/tex]
3. Now, multiply these terms together:
- [tex]\((\operatorname{cosec} \theta - \sin \theta) (\sec \theta - \cos \theta) (\tan \theta + \cot \theta) \)[/tex]
- [tex]\( = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) (2) \)[/tex]
- [tex]\( = \left(\frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2}\right) \times 2 \)[/tex]
- [tex]\( = \left(\frac{2}{4}\right) \times 2 \)[/tex]
- [tex]\( = \frac{1}{2} \times 2 = 1 \)[/tex]
### Final Result
Thus, the value of the expression [tex]\((\operatorname{cosec} \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta)\)[/tex] when [tex]\(\theta = \frac{\pi}{4}\)[/tex] is [tex]\(1\)[/tex].
