Use identities to simplify each expression.
sin(x)+cos^2(x)/sin(x) = ?
tan^3(x)−sec^2(x)tan(x)/cot(−x) = ?
sin^4(x)−cos^4(x) =?
sin + cos^2/sin
= sin + (1-sin^2)/sin
= sin + 1/sin - sin^2/sin
= sin + csc - sin
= csc
tan^3 - sec^2tan/cot(-x)
= tan^3 + sec^2tan^2
= tan^3 + (tan^2+1)tan^2
= tan^4 + tan^3 + tan^2
= tan^2(tan^2+tan+1)
not sure where you want to go with this one
sin^4-cos^4
= (sin^2+cos^2)(sin^2-cos^2)
= -cos(2x)
To simplify the given expressions, let's use trigonometric identities.
1. sin(x) + cos^2(x)/sin(x):
We can simplify this expression by using the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Rearranging the Pythagorean identity, we have: sin^2(x) = 1 - cos^2(x).
Substituting this into our original expression, we get:
sin(x) + cos^2(x)/sin(x) = sin(x) + (1 - cos^2(x))/sin(x).
To simplify the expression further, we can divide the terms individually:
sin(x)/sin(x) + (1 - cos^2(x))/sin(x) = 1 + (1 - cos^2(x))/sin(x).
Now, let's simplify the fraction: (1 - cos^2(x))/sin(x).
Using the Pythagorean identity again, we know that sin^2(x) = 1 - cos^2(x), which means cos^2(x) = 1 - sin^2(x).
Substituting this into the fraction above, we get: (1 - (1 - sin^2(x)))/sin(x).
Simplifying further, we have: (1 - 1 + sin^2(x))/sin(x) = sin^2(x)/sin(x) = sin(x).
Finally, substituting sin(x) back into the original expression, we have:
1 + sin(x) = sin(x) + 1.
Therefore, sin(x) + cos^2(x)/sin(x) simplifies to sin(x) + 1.
2. tan^3(x) − sec^2(x)tan(x)/cot(−x):
Let's use the following trigonometric identities to simplify this expression:
- tan(x) = sin(x)/cos(x)
- sec(x) = 1/cos(x)
- cot(x) = cos(x)/sin(x)
Substituting these identities into our original expression, we have:
(sin(x)/cos(x))^3 - (1/cos^2(x))(sin(x)/cos(x))(cos(x)/(-sin(x))).
Simplifying further, we get:
sin^3(x)/cos^3(x) - (sin(x)/cos^3(x))(-1/sin(x)).
Now, let's simplify the terms individually:
sin^3(x)/cos^3(x) = (sin(x)/cos(x))^3 = tan^3(x).
-(sin(x)/cos^3(x))(-1/sin(x)) = (sin(x)/cos(x))^3 = tan^3(x).
Therefore, tan^3(x) - sec^2(x)tan(x)/cot(−x) simplifies to tan^3(x) - tan^3(x).
Subtracting tan^3(x) from tan^3(x), we get 0 as the answer.
3. sin^4(x) - cos^4(x):
We can use the difference of squares identity to simplify this expression:
a^2 - b^2 = (a + b)(a - b).
Applying the difference of squares identity, we have:
sin^4(x) - cos^4(x) = (sin^2(x) + cos^2(x))(sin^2(x) - cos^2(x)).
Since sin^2(x) + cos^2(x) = 1 (based on the Pythagorean identity), our expression simplifies to:
1(sin^2(x) - cos^2(x)).
Again, using the Pythagorean identity sin^2(x) = 1 - cos^2(x), we can substitute this value into our expression:
1(1 - cos^2(x) - cos^2(x)) = 1(1 - 2cos^2(x)) = 1 - 2cos^2(x).
Therefore, sin^4(x) - cos^4(x) simplifies to 1 - 2cos^2(x).