simplify to a constant or trig func.
1. sec ²u-tan ²u/cos ²v+sin ²v
change expression to only sines and cosines. then to a basic trig function.
2. sin(theta) - tan(theta)*cos(theta)+ cos(pi/2 - theta)
3. (sec y - tan y)(sec y + tan y)/ sec y
combine fractions and simplify to a multiple of a power of a basic trig func.
4. (1/(1-sin x)) + ((1/(1+sin x))
5. sin x/1-cos x + 1-cos x/ sin x
write in factoed form of a single trig func.
6. 1-2sin x + sin²x
7. sin x - cos²x - 1
1. To simplify the expression, we'll start by using the Pythagorean identity: sin^2(u) + cos^2(u) = 1.
Sec^2(u) - tan^2(u) / cos^2(v) + sin^2(v)
Using the identity tan^2(u) = sec^2(u) - 1:
Sec^2(u) - (sec^2(u) - 1) / cos^2(v) + sin^2(v)
Combining like terms:
1 / cos^2(v) + sin^2(v)
Using the Pythagorean identity again, we have:
1 / (cos^2(v) + sin^2(v))
This simplifies to:
1 / 1
Which is just a constant.
2. Simplifying the expression step-by-step:
sin(theta) - tan(theta)*cos(theta) + cos(pi/2 - theta)
Using the identity tan(theta) = sin(theta) / cos(theta):
sin(theta) - (sin(theta) / cos(theta)) * cos(theta) + cos(pi/2 - theta)
Simplifying further:
sin(theta) - sin(theta) + cos(pi/2 - theta)
Using the identity sin(pi/2 - theta) = cos(theta):
cos(theta)
The expression simplifies to a basic trigonometric function, cos(theta).
3. Simplifying the expression step-by-step:
(sec(y) - tan(y))(sec(y) + tan(y)) / sec(y)
Using the identity sec^2(y) - tan^2(y) = 1:
(1)(sec(y) + tan(y)) / sec(y)
Canceling out sec(y):
sec(y) + tan(y)
The expression simplifies to sec(y) + tan(y), which is a multiple of a power of a basic trigonometric function.
4. Simplifying the expression step-by-step:
(1/(1 - sin(x))) + (1/(1 + sin(x)))
Using the common denominator (1 - sin(x))(1 + sin(x)):
(1 + 1 - sin(x) + 1) / (1 - sin(x))(1 + sin(x))
Simplifying further:
(3 - sin(x)) / (1 - sin^2(x))
Using the Pythagorean identity sin^2(x) = 1 - cos^2(x):
(3 - sin(x)) / (1 - (1 - cos^2(x)))
Simplifying more:
(3 - sin(x)) / cos^2(x)
The expression simplifies to a multiple of a power of a basic trigonometric function, (3 - sin(x)) / cos^2(x).
5. Simplifying the expression step-by-step:
sin(x) / (1 - cos(x)) + (1 - cos(x)) / sin(x)
Using the common denominator sin(x)(1 - cos(x)):
sin^2(x) / sin(x)(1 - cos(x)) + (1 - cos(x))(1 - cos(x)) / sin(x)(1 - cos(x))
Simplifying further:
sin^2(x) + (1 - cos(x))^2 / sin(x)(1 - cos(x))
Expanding (1 - cos(x))^2:
sin^2(x) + (1 - 2cos(x) + cos^2(x)) / sin(x)(1 - cos(x))
Simplifying more:
sin^2(x) + 1 - 2cos(x) + cos^2(x) / sin(x)(1 - cos(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:
1 + 1 - 2cos(x) / sin(x)(1 - cos(x))
Simplifying further:
2 - 2cos(x) / sin(x)(1 - cos(x))
Factoring out 2:
2(1 - cos(x)) / sin(x)(1 - cos(x))
Canceling out (1 - cos(x)):
2 / sin(x)
The expression simplifies to a multiple of a power of a basic trigonometric function, 2 / sin(x).
6. Simplifying the expression step-by-step:
1 - 2sin(x) + sin^2(x)
Rearranging the terms:
1 + sin^2(x) - 2sin(x)
Using the identity sin^2(x) = 1 - cos^2(x):
1 + (1 - cos^2(x)) - 2sin(x)
Simplifying further:
2 - cos^2(x) - 2sin(x)
The expression can be simplified no further while maintaining the same form.
7. Simplifying the expression step-by-step:
sin(x) - cos^2(x) - 1
Using the identity cos^2(x) = 1 - sin^2(x):
sin(x) - (1 - sin^2(x)) - 1
Simplifying further:
sin(x) - 1 + sin^2(x) - 1
Combining like terms:
sin^2(x) + sin(x) - 2
The expression can be simplified no further while maintaining the same form.
To simplify these expressions, we will use some trigonometric identities and properties. Let's break down each expression step by step:
1. sec²u - tan²u / cos²v + sin²v
To write the expression using only sines and cosines, we can use the following identities:
- sec²u = 1/cos²u
- tan²u = sin²u/cos²u
Substituting these identities into the expression, we have:
(1/cos²u - sin²u/cos²u) / cos²v + sin²v
=> (1 - sin²u) / cos²u / cos²v + sin²v
=> cos²u / cos²u(cos²v + sin²v) - sin²u / cos²u(cos²v + sin²v)
=> cos²u - sin²u / cos²v + sin²v
Now, to simplify this expression to a basic trigonometric function, we can use the identity:
- cos²x - sin²x = cos(2x)
Therefore, the simplified expression is:
cos(2u) / (cos²v + sin²v)
2. sin(theta) - tan(theta) * cos(theta) + cos(pi/2 - theta)
Let's work on simplifying each term in the expression:
sin(theta) - tan(theta) * cos(theta) + cos(pi/2 - theta)
=> sin(theta) - (sin(theta) / cos(theta)) * cos(theta) + cos(pi/2 - theta)
By simplifying the expression within the parentheses, we have:
sin(theta) - sin(theta) + cos(pi/2 - theta)
=> 0 + cos(pi/2 - theta)
Using the identity:
cos(pi/2 - theta) = sin(theta)
The simplified expression is:
sin(theta)
3. (sec y - tan y)(sec y + tan y) / sec y
In order to simplify this expression, let's expand the numerator and see if there are any terms that can be canceled out:
(sec y - tan y)(sec y + tan y) = sec²y - tan²y
Using the identity:
sec²y - tan²y = 1
Thus, the expression simplifies to:
1 / sec y
Using the reciprocal identity:
1 / sec y = cos y
Therefore, (sec y - tan y)(sec y + tan y) / sec y simplifies to:
cos y
4. (1/(1-sin x)) + (1/(1+sin x))
To simplify this expression, let's find a common denominator:
(1/(1-sin x)) + (1/(1+sin x))
= (1 + 1 - sin x) / (1 - sin²x)
= (2 - sin x) / cos²x
Using the identity:
1 - sin²x = cos²x,
The expression simplifies to:
(2 - sin x) / cos²x
5. sin x / (1 - cos x) + (1 - cos x) / sin x
To simplify this expression, let's find a common denominator:
sin x / (1 - cos x) + (1 - cos x) / sin x
= (sin²x + (1 - cos x)²) / (sin x)(1 - cos x)
= (sin²x + (1 - 2cos x + cos²x)) / (sin x)(1 - cos x)
= (1 - cos x + sin²x) / (sin x)(1 - cos x)
Using the identity:
1 - cos²x = sin²x,
The expression simplifies to:
1 / sin x = csc x
6. 1 - 2sin x + sin²x
This expression is already in simplified form.
7. sin x - cos²x - 1
Let's rewrite this expression using the identity:
cos²x = 1 - sin²x
sin x - cos²x - 1 =
sin x - (1 - sin²x) - 1 =
sin x - 1 + sin²x - 1 =
sin x - 2 + sin²x
This is the simplified form of the expression.