Question

Помогите упростить выражения: 1. 1-cos^2a + sin^2a 2. sina / 1+cosa + 1+cosa/sina 3. (1/cos^2a-1) ctg^2a 4. sin^3a + cos^3a/ 1-sina cosa - cosa) ctga

Answer 1

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Answer 2

1. 1 - cos^2a + sin^2a: Using the identity sin^2a + cos^2a = 1, we can simplify the expression: 1 - cos^2a + sin^2a = sin^2a + cos^2a - cos^2a + sin^2a = 2sin^2a

2. sina / (1 + cosa) + (1 + cosa) / sina: To simplify this expression, let's find a common denominator: sina * sina / (sina * (1 + cosa)) + (1 + cosa) * (1 + cosa) / (sina * (1 + cosa)) = sina^2 / (sina + sina * cosa) + (1 + 2cosa + cosa^2) / (sina + sina * cosa) Now, we can combine the fractions: = (sina^2 + 1 + 2cosa + cosa^2) / (sina + sina * cosa)

3. (1 / cos^2a - 1) * ctg^2a: Using the identity ctg^2a = 1 / tan^2a and 1 / cos^2a = sec^2a, we can simplify the expression: (1 / cos^2a - 1) * ctg^2a = (sec^2a - 1) * (1 / tan^2a) = sec^2a / tan^2a - 1 / tan^2a = (1 / sin^2a) - (1 / cos^2a)

4. (sin^3a + cos^3a) / (1 - sina cosa - cosa) * ctga: To simplify this expression, let's factor the numerator and denominator: (sin^3a + cos^3a) / (1 - sina cosa - cosa) * ctga = [(sin^3a + cos^3a) / (1 - sina cosa - cosa)] * ctga = [(sin^3a + cos^3a) / (-sina cosa + 1 - cosa)] * ctga Now, let's simplify the numerator: sin^3a + cos^3a = (sin^2a + sinacosa + cos^2a)(sin^2a - sinacosa + cos^2a) Using the identity sin^2a + cos^2a = 1, we can further simplify: = (1 + sinacosa)(1 - sinacosa + 1) = (2 + sinacosa - sinacosa^2) Finally, we substitute this back into the expression: [(2 + sinacosa - sinacosa^2) / (-sina cosa + 1 - cosa)] * ctga