Simplify each expression using the fundamental identities.
1. Sec x csc x / sec^2x+csc^2x
2. Sec x + tan x / sec x + tan x - cos x
sin^2+cos^2 = 1, so
sec^2 + csc^2 = 1/cos^2 + 1/sin^2
= (sin^2+cos^2)/(sin^2 cos^2)
= 1/(sin^2 cos^2)
= sec^2 csc^2
so,
(sec csc) / (sec^2 csc^2)
= 1/(sec csc)
= sin cos
= 1/2 sin(2x)
#2 is missing too many parentheses to work on it. That dangling cos x bothers me.
To simplify the expressions using the fundamental identities, we need to recognize the trigonometric identities that involve the functions in the given expressions. Here are the steps to simplify each expression:
1. Sec x csc x / (sec^2x + csc^2x):
Start by substituting the reciprocal identities for sec x and csc x:
(sec x * csc x) / (1/cos^2x + 1/sin^2x)
Next, simplify the denominators:
(sec x * csc x) / ((sin^2x + cos^2x) / (cos^2x * sin^2x))
Combine the fractions by multiplying the reciprocal of the denominator:
(sec x * csc x) * (cos^2x * sin^2x) / (sin^2x + cos^2x)
Apply the Pythagorean Identity (sin^2x + cos^2x = 1):
(sec x * csc x) * (cos^2x * sin^2x) / 1
Simplify by canceling out the common factors:
cos^2x * sin^2x / 1
The final simplified expression is: cos^2x * sin^2x.
2. (Sec x + tan x) / (sec x + tan x - cos x):
Start by factoring out sec x from the numerator and denominator:
sec x * (1 + sin x) / (sec x * (1 - cos x))
Cancel out the common factor sec x:
(1 + sin x) / (1 - cos x)
To simplify further, we can rationalize the denominator:
[(1 + sin x) * (1 + cos x)] / [(1 - cos x) * (1 + cos x)]
Expand and simplify the numerator and denominator:
(1 + sin x + cos x + sin x * cos x) / (1 - cos^2x)
Apply the Pythagorean Identity (cos^2x = 1 - sin^2x) to the denominator:
(1 + sin x + cos x + sin x * cos x) / sin^2x
Rearrange the terms in the numerator:
(1 + cos x + sin x + sin x * cos x) / sin^2x
Factor out a sin x from the numerator:
(1 + sin x) * (1 + cos x) / sin^2x
The final simplified expression is: (1 + sin x) * (1 + cos x) / sin^2x.