How to do this one?
(sina/1-cosa) + (1-cosa/sina) =
Also, wanted to ask if I did another one right:
1 - sin^2a + tg^2a * cos^2a = 1 - sin^2a + (sina/cosa)^2 * cos^2a = 1 - sin^2a + sin^2a = 1 ???
That's not what I get.
(1-cosa)/sina = tan(a/2)
so, you have
1/tan(a/2) + tan(a/2)
= (1 + tan^2(a/2))/tan(a/2)
= sec^2(a/2)/tan(a/2)
= 1/cos^2(a/2) * cos(a/2)/sin(a/2)
= 1/sin(a/2)cos(a/2) = 2/sina
Well it goes like this :
sinA/1-cosA + 1-cosA/sinA
= sin^2A + (1-cosA)^2/(1-cosA)sinA (cross multiplying)
= sin^2A + cos^2A + 1 - 2cosA/(1-cosA)sinA
= 1 + 1 - 2cosA/(1-cosA)sinA
(sin^2A + cos^2A =1)
= 2 - 2cosA/(1-cosA)sinA
= 2 (1-cosA)/(1-cosA)sinA
= 2/sinA
To solve the first equation:
Step 1: Get a common denominator:
Multiply the first fraction by (1 - cosa) / (1 - cosa), and the second fraction by sina/sina to get a common denominator of sina(1 - cosa):
[(sina(1 - cosa)) / (1 - cosa)] + [(1 - cosa ) / sina]
= [sina(1 - cosa) + (1 - cosa)] / sina(1 - cosa)
Step 2: Simplify the numerator:
sina - sina * cosa + 1 - cosa
Combine like terms:
1 - sina * cosa - sina * cosa
= 1 - 2sina * cosa
Step 3: Simplify the expression:
Final result: (1 - 2sina * cosa) / sina(1 - cosa)
Now, let's check your second equation:
1 - sin^2a + tg^2a * cos^2a = 1 - sin^2a + (sina/cosa)^2 * cos^2a
Step 1: Simplify the expressions:
Using the trigonometric identity tg^2a = (sina/cosa)^2:
1 - sin^2a + (sina/cosa)^2 * cos^2a
= 1 - sin^2a + (sina^2/cosa^2) * cos^2a
= 1 - sin^2a + (sina^2/cosa^2) * (1 - sin^2a)
= 1 - sin^2a + (sina^2 * (1 - sin^2a)) / cosa^2
Step 2: Distribute and simplify:
= 1 - sin^2a + (sina^2 - sina^4) / cosa^2
= 1 - sin^2a + sina^2 / cosa^2 - sina^4 / cosa^2
Step 3: Combine like terms:
= 1 + sina^2 / cosa^2 - sin^2a - sina^4 / cosa^2
Step 4: Combine the terms with a common denominator:
= (cosa^2 + sina^2 - sin^2a - sina^4) / cosa^2
Step 5: Simplify the numerator:
Since cosa^2 + sina^2 = 1 (by the Pythagorean identity), the numerator simplifies to:
= (1 - sin^2a - sina^4) / cosa^2
= 1 - sin^2a - sina^4 / cosa^2
= 1 - sin^2a - (sina^2)^2 / cosa^2
= 1 - sin^2a - sin^4a / cosa^2
Step 6: Combine like terms:
Final result: 1 - sin^2a - sin^4a / cosa^2
So, your calculation is incorrect. The correct simplified result is 1 - sin^2a - sin^4a / cosa^2.