Use fundamental identities to simplify the expression
sinx+1 + 1/sinx-1
surely you must mean
sinx + 1 + 1/(six- 1)
if so , then
the LCD is sinx - 1
and we get
( (sinx +1)(sinx - 1) +1) )/(sinx - 1)
= sin^2 x - 1+ 1)/(sinx + 1)
= sin^2 x /(sinx + 1)
To simplify the expression sin(x)+1 + 1/sin(x)-1 using fundamental trigonometric identities, let's first find a common denominator. We'll multiply the first term (sin(x)+1) by (sin(x)-1)/(sin(x)-1) and multiply the second term (1/sin(x)-1) by (1+sin(x))/(1+sin(x)):
(sin(x)+1)(sin(x)-1)/(sin(x)-1) + (1/sin(x)-1)(1+sin(x))/(1+sin(x))
Expanding and simplifying each term:
(sin^2(x)-1)/(sin(x)-1) + (1+sin(x))/(sin(x)(1+sin(x))-1)
Now, let's simplify further using the identity sin^2(x) + cos^2(x) = 1:
(cos^2(x))/(sin(x)-1) + (1+sin(x))/(sin(x)(1+sin(x))-1)
Next, let's simplify the denominator by factoring out a common factor of sin(x):
(cos^2(x))/(sin(x)-1) + (1+sin(x))/(sin(x)((1+sin(x))/sin(x))-1)
(cos^2(x))/(sin(x)-1) + (1+sin(x))/(1+sin(x))-1
Now, we can simplify even further:
(cos^2(x))/(sin(x)-1) + 1
Finally, we can use the identity 1 - sin^2(x) = cos^2(x) to rewrite the numerator:
[1 - sin^2(x)]/(sin(x)-1) + 1
Now, we have simplified the expression using fundamental trigonometric identities.