How do you simplify:
(1/(sin^2x-cos^2x))-(2/cosx-sinx)?
I tried factoring and creating a LCD of (sinx+cosx)(sinx cosx) (cosx-sinx), but cannot come up with the right answer.
The answer is (1+2sinx+2cosx)/sin^2x+cos^2x,but I don't know how the book arrived at that answer. I'd appreciate any help. Please explain your answer. Thank you.
1/(sin^2x-cos^2x) = 1/[(sinx-cosx)(sinx+cosx)]
so, using a common denominator,
1/(sin^2x-cos^2x) - 2/(cosx-sinx)
= 1/[(sinx-cosx)(sinx+cosx)] + 2(sinx+cosx)/[(sinx-cosx)(sinx+cosx)]
= (1+2sinx+2cosx)/(sin^2x-cos^2x)
I see several typos in your posting. If I have interpreted things wrong, feel free to fix it, using extra parentheses if needed...
Factoring looks promising:
(1/(sin^2x-cos^2x))-(2/cosx-sinx)
= 1/( (sinx+cosx)(sinx-cosx) ) + 2/(sinx-cosx)
so (sinx+cosx)(sinx-cosx) is the LCD
= 1/( (sinx+cosx)(sinx-cosx) ) + 2(sinx+cosx)/( (sinx+cosx)(sinx-cosx) )
= ( 1 + 2sinx + 2cosx)/(sin^2 x - cos^2 x)
You have a typo in the answer you posted. Your denominator of the answer would be just 1.
To simplify the expression (1/(sin^2x-cos^2x)) - (2/cosx-sinx), let's start by factoring the denominators:
sin^2x - cos^2x can be factored using the difference of two squares formula as (sinx - cosx)(sinx + cosx),
cosx - sinx can be written as -(sinx - cosx) to match the factored form of the first denominator.
The expression now becomes:
1/((sinx - cosx)(sinx + cosx)) - 2/(-(sinx - cosx)).
Next, let's simplify the expression by finding a common denominator. The common denominator will be (sinx - cosx)(sinx + cosx), which can be achieved by multiplying the first term's denominator by -(sinx - cosx):
1/((sinx - cosx)(sinx + cosx)) - 2(sinx + cosx)/((sinx - cosx)(sinx + cosx)).
Combining the fractions over the common denominator, we have:
(1 - 2(sinx + cosx))/((sinx - cosx)(sinx + cosx)).
Now, let's simplify the numerator:
1 - 2(sinx + cosx) = 1 - 2sinx - 2cosx.
Substituting it back, the expression becomes:
(1 - 2sinx - 2cosx)/((sinx - cosx)(sinx + cosx)).
We can further simplify the numerator:
1 - 2sinx - 2cosx can be written as (1 - 2cosx - 2sinx), rearranging the terms:
(1 + 2sinx + 2cosx)/((sinx - cosx)(sinx + cosx)).
Lastly, since sin^2x + cos^2x equals 1 (using the Pythagorean identity), we can simplify the denominator:
(sinx - cosx)(sinx + cosx) equals sin^2x - cos^2x, which is equivalent to 1.
Thus, the final simplified expression is:
(1 + 2sinx + 2cosx)/(sin^2x + cos^2x), which further simplifies to:
(1 + 2sinx + 2cosx)/(1), or simply:
1 + 2sinx + 2cosx.
Therefore, the answer is (1 + 2sinx + 2cosx).