Simplify and write the trigonometric expression in terms of sine and cosine:
((2+tan^2 x)/(sec^2 x))-1 = (f(x))^2
f(x)= ??
To simplify and write the trigonometric expression in terms of sine and cosine, let's break down each term:
1. Start by simplifying the numerator of the fraction by using the identity tan^2(x) = sec^2(x) - 1:
((2 + tan^2(x))/(sec^2(x))) - 1 = ((2 + (sec^2(x) - 1))/(sec^2(x))) - 1
2. Continue simplifying the numerator:
((2 + sec^2(x) - 1)/(sec^2(x))) - 1 = ((1 + sec^2(x))/(sec^2(x))) - 1
3. Simplify the denominator by using the identity sec^2(x) = 1 + tan^2(x):
((1 + sec^2(x))/(1 + tan^2(x))) - 1 = ((1 + (1 + tan^2(x)))/(1 + tan^2(x))) - 1
4. Combine like terms in the numerator:
(((2 + tan^2(x) + 1))/(1 + tan^2(x))) - 1 = ((3 + tan^2(x))/(1 + tan^2(x))) - 1
Now, let's write the expression in terms of sine and cosine. Recall that tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x):
5. Replace tan^2(x) with (sin^2(x)/cos^2(x)) and sec^2(x) with (1/cos^2(x)) to get:
((3 + (sin^2(x)/cos^2(x)))/(1 + (sin^2(x)/cos^2(x)))) - 1
6. To combine the fractions, find a common denominator of cos^2(x):
((3cos^2(x) + sin^2(x))/(cos^2(x) + sin^2(x))) - 1
7. Simplify further using the identity cos^2(x) + sin^2(x) = 1:
(3cos^2(x) + sin^2(x))/1 - 1 = 3cos^2(x) + sin^2(x) - 1
Therefore, the simplified expression in terms of sine and cosine is:
f(x) = √(3cos^2(x) + sin^2(x) - 1)