use the fundamental trig. Identities to simplify tan^4x+2tan^2x+1
tan^4x+2tan^2x+1
= (tan^2x+ 1)(tan^2x+ 1) = (tan^2x+1)^2
but we know tan^2x + 1 = sec^2x
so above
= sec^4x
To simplify the expression tan^4(x) + 2tan^2(x) + 1 using the fundamental trigonometric identities, we can start by rewriting the expression in terms of sin(x) and cos(x).
Recall that tan(x) is equal to sin(x) / cos(x). Therefore, we can express tan^2(x) as (sin(x) / cos(x))^2, which simplifies to sin^2(x) / cos^2(x).
Now, we can rewrite the given expression as follows:
tan^4(x) + 2tan^2(x) + 1
= (sin^2(x) / cos^2(x))^2 + 2(sin^2(x) / cos^2(x)) + 1
Next, let's simplify the expression using the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Since sin^2(x) = 1 - cos^2(x), we can substitute this value in the expression:
= ((1 - cos^2(x)) / cos^2(x))^2 + 2((1 - cos^2(x)) / cos^2(x)) + 1
Expanding and simplifying the expression further:
= (1 - 2cos^2(x) + cos^4(x)) / cos^4(x) + (2 - 2cos^2(x)) / cos^2(x) + 1
= 1/cos^4(x) - 2cos^2(x)/cos^4(x) + cos^4(x)/cos^4(x)
+ 2/cos^2(x) - 2cos^2(x)/cos^2(x) + 1
Now, we can combine like terms and simplify the expression:
= 1/cos^4(x) - 2cos^2(x)/cos^4(x) + 1 + 2/cos^2(x) - 2 + 1
= 1/cos^4(x) - 2cos^2(x)/cos^4(x) + 3/cos^2(x)
Finally, we can combine the fractions:
= (1 - 2cos^2(x) + 3cos^2(x)) / cos^4(x)
= (1 + cos^2(x)) / cos^4(x)
Therefore, the simplified expression is (1 + cos^2(x)) / cos^4(x).
To simplify the expression tan^4x + 2tan^2x + 1 using the fundamental trigonometric identities, we can rewrite the given expression in terms of sine and cosine functions.
First, recall the fundamental trigonometric identity:
tan^2x = sec^2x - 1
Using this identity, we can rewrite the expression as follows:
tan^4x + 2tan^2x + 1
= (tan^2x)^2 + 2tan^2x + 1
= (sec^2x - 1)^2 + 2(sec^2x - 1) + 1
= sec^4x - 2sec^2x + 1 + 2sec^2x - 2 + 1
= sec^4x + sec^2x
Note that sec^2x is the reciprocal of cos^2x, so we can express the simplified expression entirely in terms of cosine (cos) as follows:
sec^4x + sec^2x
= (1/cos^2x)^2 + 1/cos^2x
= 1/cos^4x + 1/cos^2x
Thus, the simplified expression in terms of cosine is 1/cos^4x + 1/cos^2x.