I'm trying to see if the teacher didn't give us the right problem or if I'm just not getting it. The problem is "re-write the expression without a fraction: 4/(1-sin x)". I can do it if the problem is sin squared x but I don't know how to solve the problem given. Pls help. thx
multiply by (1+sinx)/(1+sinx) which is really 1, so you are not changing the value, only the appearance.
4/(1-sin x)[(1+sinx)/(1+sinx)] to get
4(1+sinx)/(1 - sin^2 x)
= 4(1+sinx)/cos^2 x
= 4(sec^2 x)(1 + sinx)
To rewrite the expression without a fraction, you need to eliminate the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of 1-sin(x) is 1+sin(x).
So, to rewrite the expression 4/(1-sin(x)) without a fraction, you can multiply both the numerator and denominator by (1+sin(x)):
4/(1-sin(x)) * (1+sin(x))/(1+sin(x))
Now, let's simplify the expression step by step:
Numerator: 4 * (1 + sin(x)) = 4 + 4sin(x)
Denominator: (1 - sin(x)) * (1 + sin(x)) = 1 - sin^2(x) (using the difference of squares formula)
Since sin^2(x) is equal to 1 - cos^2(x), you can substitute it in the denominator:
1 - sin^2(x) = 1 - (1 - cos^2(x)) = cos^2(x)
So, the simplified expression is:
(4 + 4sin(x))/cos^2(x)
Therefore, the expression 4/(1-sin(x)) can be rewritten without a fraction as (4 + 4sin(x))/cos^2(x).