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Sagot :
Sure, let's tackle the simplification of the given expressions and then verify the given relationship for the angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex].
### Simplification of Expressions
#### a) Simplify [tex]\( \sum_{t=0}^1 (t+1) \)[/tex]:
The sum of the series [tex]\( \sum_{t=0}^1 (t+1) \)[/tex] can be evaluated by substituting the values of [tex]\( t \)[/tex] within the specified range:
[tex]\[ \sum_{t=0}^1 (t+1) = (0+1) + (1+1) = 1 + 2 = 3 \][/tex]
Therefore, the simplified expression is 3.
#### b) Simplify [tex]\( \frac{\cos^2 \theta}{1 - \sin \theta} \)[/tex]:
To simplify [tex]\( \frac{\cos^2 \theta}{1 - \sin \theta} \)[/tex], we can use the Pythagorean identity, which states [tex]\( \cos^2 \theta = 1 - \sin^2 \theta \)[/tex].
[tex]\[ \frac{\cos^2 \theta}{1 - \sin \theta} = \frac{1 - \sin^2 \theta}{1 - \sin \theta} \][/tex]
Notice that the numerator [tex]\( 1 - \sin^2 \theta \)[/tex] can be factored as [tex]\( (1 - \sin \theta)(1 + \sin \theta) \)[/tex]:
[tex]\[ \frac{1 - \sin^2 \theta}{1 - \sin \theta} = \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 - \sin \theta} \][/tex]
Since [tex]\( 1 - \sin \theta \)[/tex] is in both the numerator and the denominator, we can cancel them:
[tex]\[ \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 - \sin \theta} = 1 + \sin \theta \][/tex]
Thus, the simplified expression is [tex]\( 1 + \sin \theta \)[/tex].
#### c) Simplify [tex]\( \sin^3 \alpha + \sin \alpha \cos^2 \alpha \)[/tex]:
To simplify [tex]\( \sin^3 \alpha + \sin \alpha \cos^2 \alpha \)[/tex], we can factor out [tex]\( \sin \alpha \)[/tex]:
[tex]\[ \sin^3 \alpha + \sin \alpha \cos^2 \alpha = \sin \alpha (\sin^2 \alpha + \cos^2 \alpha) \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \alpha + \cos^2 \alpha = 1 \)[/tex]:
[tex]\[ \sin \alpha (\sin^2 \alpha + \cos^2 \alpha) = \sin \alpha \cdot 1 = \sin \alpha \][/tex]
Hence, the simplified expression is [tex]\( \sin \alpha \)[/tex].
### Verifying the Relationship for Angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
For this part, let's examine what the relationship is supposed to be. Since the original question seems to only indicate that there is a relationship without specifying it, let's assume it’s a well-known trigonometric identity involving angles. One common relationship is the angle sum and difference identity:
For example, a common identity to verify might be [tex]\( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)[/tex].
However, since the specific relationship is not provided, we assume that you have a particular relationship in mind between angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]. Please provide the specific relationship to be verified so that I can complete this part accurately.
If you need any further clarification or have another relationship involving angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex], feel free to provide it, and I will be glad to verify it for you.
### Simplification of Expressions
#### a) Simplify [tex]\( \sum_{t=0}^1 (t+1) \)[/tex]:
The sum of the series [tex]\( \sum_{t=0}^1 (t+1) \)[/tex] can be evaluated by substituting the values of [tex]\( t \)[/tex] within the specified range:
[tex]\[ \sum_{t=0}^1 (t+1) = (0+1) + (1+1) = 1 + 2 = 3 \][/tex]
Therefore, the simplified expression is 3.
#### b) Simplify [tex]\( \frac{\cos^2 \theta}{1 - \sin \theta} \)[/tex]:
To simplify [tex]\( \frac{\cos^2 \theta}{1 - \sin \theta} \)[/tex], we can use the Pythagorean identity, which states [tex]\( \cos^2 \theta = 1 - \sin^2 \theta \)[/tex].
[tex]\[ \frac{\cos^2 \theta}{1 - \sin \theta} = \frac{1 - \sin^2 \theta}{1 - \sin \theta} \][/tex]
Notice that the numerator [tex]\( 1 - \sin^2 \theta \)[/tex] can be factored as [tex]\( (1 - \sin \theta)(1 + \sin \theta) \)[/tex]:
[tex]\[ \frac{1 - \sin^2 \theta}{1 - \sin \theta} = \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 - \sin \theta} \][/tex]
Since [tex]\( 1 - \sin \theta \)[/tex] is in both the numerator and the denominator, we can cancel them:
[tex]\[ \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 - \sin \theta} = 1 + \sin \theta \][/tex]
Thus, the simplified expression is [tex]\( 1 + \sin \theta \)[/tex].
#### c) Simplify [tex]\( \sin^3 \alpha + \sin \alpha \cos^2 \alpha \)[/tex]:
To simplify [tex]\( \sin^3 \alpha + \sin \alpha \cos^2 \alpha \)[/tex], we can factor out [tex]\( \sin \alpha \)[/tex]:
[tex]\[ \sin^3 \alpha + \sin \alpha \cos^2 \alpha = \sin \alpha (\sin^2 \alpha + \cos^2 \alpha) \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \alpha + \cos^2 \alpha = 1 \)[/tex]:
[tex]\[ \sin \alpha (\sin^2 \alpha + \cos^2 \alpha) = \sin \alpha \cdot 1 = \sin \alpha \][/tex]
Hence, the simplified expression is [tex]\( \sin \alpha \)[/tex].
### Verifying the Relationship for Angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
For this part, let's examine what the relationship is supposed to be. Since the original question seems to only indicate that there is a relationship without specifying it, let's assume it’s a well-known trigonometric identity involving angles. One common relationship is the angle sum and difference identity:
For example, a common identity to verify might be [tex]\( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)[/tex].
However, since the specific relationship is not provided, we assume that you have a particular relationship in mind between angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]. Please provide the specific relationship to be verified so that I can complete this part accurately.
If you need any further clarification or have another relationship involving angles [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex], feel free to provide it, and I will be glad to verify it for you.
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