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Sagot :
To analyze the given trigonometric expression and verify whether it simplifies to the target expression [tex]\(\tan \theta \cdot \sin \theta\)[/tex], let's break down each part step by step.
Given expression:
[tex]\[ \frac{\sin^2 \theta}{\sin^2 (90^\circ - \theta)} \cdot \frac{\sin (90^\circ - \theta)}{\tan \theta} \cdot \cot (90^\circ - \theta) \][/tex]
### Step 1: Simplifying [tex]\(\sin (90^\circ - \theta)\)[/tex] and [tex]\(\cot (90^\circ - \theta)\)[/tex]
We know that:
[tex]\[ \sin (90^\circ - \theta) = \cos \theta \][/tex]
[tex]\[ \cot (90^\circ - \theta) = \tan \theta \][/tex]
### Step 2: Substitute these identities into the expression
Using these identities, the expression becomes:
[tex]\[ \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos \theta}{\tan \theta} \cdot \tan \theta \][/tex]
### Step 3: Simplifying [tex]\(\tan \theta\)[/tex]
Recall that:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Thus:
[tex]\[ \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \][/tex]
### Step 4: Substitute [tex]\(\tan\)[/tex] identities in the expression
Now substitute [tex]\(\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)[/tex] back into the expression:
[tex]\[ \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \cos \theta \cdot \frac{\cos \theta}{\sin \theta} \cdot \tan \theta \][/tex]
### Step 5: Simplify the expression
Simplify each component of the expression:
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right) \cdot \left( \cos \theta \right) \cdot \left( \frac{\cos \theta}{\sin \theta} \right) \cdot \left( \frac{\sin \theta}{\cos \theta} \right) \][/tex]
Combine the terms and simplify fractions:
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right) \cdot \left( \frac{\cos^2 \theta}{\sin \theta} \right) \cdot \sin \theta \][/tex]
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos^2 \theta}{\sin^2 \theta} \right) \cdot \sin \theta \][/tex]
[tex]\[ = 1 \cdot \sin \theta \][/tex]
Thus:
[tex]\[ = \sin \theta \][/tex]
### Conclusion
This simplified expression is compared with the target expression:
[tex]\[ \tan \theta \cdot \sin \theta \][/tex]
Since as per the provided answer, the expressions are not equal:
[tex]\[ \left(\frac{\sin(\theta + 90) + \sin(3\theta - 90)}{2(1 - \cos(2\theta - 180))}\right) \][/tex]
So, the two expressions are not equivalent.
Thus, the given trigonometric expression does not simplify to [tex]\(\tan \theta \cdot \sin \theta\)[/tex].
Given expression:
[tex]\[ \frac{\sin^2 \theta}{\sin^2 (90^\circ - \theta)} \cdot \frac{\sin (90^\circ - \theta)}{\tan \theta} \cdot \cot (90^\circ - \theta) \][/tex]
### Step 1: Simplifying [tex]\(\sin (90^\circ - \theta)\)[/tex] and [tex]\(\cot (90^\circ - \theta)\)[/tex]
We know that:
[tex]\[ \sin (90^\circ - \theta) = \cos \theta \][/tex]
[tex]\[ \cot (90^\circ - \theta) = \tan \theta \][/tex]
### Step 2: Substitute these identities into the expression
Using these identities, the expression becomes:
[tex]\[ \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos \theta}{\tan \theta} \cdot \tan \theta \][/tex]
### Step 3: Simplifying [tex]\(\tan \theta\)[/tex]
Recall that:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Thus:
[tex]\[ \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \][/tex]
### Step 4: Substitute [tex]\(\tan\)[/tex] identities in the expression
Now substitute [tex]\(\frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)[/tex] back into the expression:
[tex]\[ \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \cos \theta \cdot \frac{\cos \theta}{\sin \theta} \cdot \tan \theta \][/tex]
### Step 5: Simplify the expression
Simplify each component of the expression:
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right) \cdot \left( \cos \theta \right) \cdot \left( \frac{\cos \theta}{\sin \theta} \right) \cdot \left( \frac{\sin \theta}{\cos \theta} \right) \][/tex]
Combine the terms and simplify fractions:
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right) \cdot \left( \frac{\cos^2 \theta}{\sin \theta} \right) \cdot \sin \theta \][/tex]
[tex]\[ = \left( \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos^2 \theta}{\sin^2 \theta} \right) \cdot \sin \theta \][/tex]
[tex]\[ = 1 \cdot \sin \theta \][/tex]
Thus:
[tex]\[ = \sin \theta \][/tex]
### Conclusion
This simplified expression is compared with the target expression:
[tex]\[ \tan \theta \cdot \sin \theta \][/tex]
Since as per the provided answer, the expressions are not equal:
[tex]\[ \left(\frac{\sin(\theta + 90) + \sin(3\theta - 90)}{2(1 - \cos(2\theta - 180))}\right) \][/tex]
So, the two expressions are not equivalent.
Thus, the given trigonometric expression does not simplify to [tex]\(\tan \theta \cdot \sin \theta\)[/tex].
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