I was given 21 questions for homework and I can't get the last few no matter how hard and how many times I try.
17. Sinx-1/sinx+1 = -cos^2x/(sinx+1)^2
18. Sin^4x + 2sin^2xcos^2x + cos^4x = 1
19. 4/cos^2x - 5 = 4tan^2x - 1
20. Cosx - sinx - cos^3x/Cosx = sin^2 - tanx
21. Sin^2x - 6sinx + 9/sin^2x - 9 = sinx-3/sinx+3
Especially the last three!
19). 4sin^2x/cos^2x - cos^2x/cos^2x
4(1-cos^2x) - cos^2x. All over cos^2x
4 - 4cos^2x - cos^2x. All over cos^2x
4 - 5cos^2x. All over cos^2x
4/cos^2x - 5
20). Sin^2x - sin x / cos x
(1-cos^2x)cos x - sin x. All over cos x
Cos x - cos^3x - sin x. All over cos x
21). Factor the numerator and the denominator
(Sin x - 3)(sin x - 3) / (sin x - 3)(sin x + 3)
17). Multiply numerator and denominator by
Sin x + 1
The numerator ( sin x + 1)(sin x -1) = sin ^2x -1
= -1(1 - sin^2x) = -cos^2x
18). Factor perfect square ( sin ^2x + cos^2x)^2
= 1^2=1
To solve these trigonometric equations, it's helpful to have a basic understanding of trigonometric identities and algebraic manipulations. I'll guide you step-by-step on how to solve each equation.
17. Sinx-1/sinx+1 = -cos^2x/(sinx+1)^2:
To simplify the left side of the equation, combine the fractions by finding a common denominator:
(sin x - 1)(sin x + 1) / (sin x + 1) = -cos^2 x / (sin x + 1)^2
Cancel out the common factor of (sin x + 1) from the numerators and denominators:
sin x - 1 = -cos^2 x / (sin x + 1)
Multiply both sides of the equation by (sin x + 1) to eliminate the denominator:
(sin x - 1)(sin x + 1) = -cos^2 x
Apply the identity sin^2 x - cos^2 x = 1:
sin^2 x - 1 = -cos^2 x
Rearrange the equation:
sin^2 x + cos^2 x = 1
Which is the Pythagorean Identity. Therefore, this equation holds true for all values of x.
18. Sin^4x + 2sin^2xcos^2x + cos^4x = 1:
Notice that this equation resembles another trigonometric identity, specifically the Pythagorean Identity (sin^2 x + cos^2 x = 1). So, let's try to use that.
Rewrite the equation as:
(sin^2 x)^2 + 2sin^2 x cos^2 x + (cos^2 x)^2 = 1
Now, substitute sin^2 x as (1 - cos^2 x):
(1 - cos^2 x)^2 + 2(1 - cos^2 x)cos^2 x + cos^4 x = 1
Expand and simplify:
1 - 2cos^2 x + cos^4 x + 2cos^2 x - 2cos^4 x + cos^4 x = 1
1 - cos^4 x = 1
The equation simplifies to 0 = 0, meaning it holds true for all values of x.
19. 4/cos^2 x - 5 = 4tan^2 x - 1:
Start by combining like terms:
4/cos^2 x - 4tan^2 x = 4
Now, let's express the right side of the equation in terms of cos and sin by using the identity tan^2 x = sin^2 x / cos^2 x:
4/cos^2 x - 4sin^2 x / cos^2 x = 4
Combine the fractions with a common denominator:
(4 - 4sin^2 x) / cos^2 x = 4
Multiply both sides of the equation by cos^2 x to eliminate the denominator:
4 - 4sin^2 x = 4cos^2 x
Rearrange the equation:
4cos^2 x + 4sin^2 x = 4
Simplify using the Pythagorean identity:
4 = 4
This equation is true for all values of x.
20. Cosx - sinx - cos^3x/Cosx = sin^2x - tanx:
To simplify the equation, let's rewrite it with a common denominator:
(cos x - sin x)(cos x) - cos^3 x / cos x = sin^2 x - tan x
Expand and combine like terms:
cos^2 x - sin x cos x - cos^3 x / cos x = sin^2 x - tan x
Rearrange the equation:
cos^2 x - sin x cos x - cos^3 x = cos x (sin^2 x - tan x)
Express sin^2 x and tan x in terms of cos x:
cos^2 x - sin x cos x - cos^3 x = cos x (1 - cos^2 x / cos x)
Simplify:
cos^2 x - sin x cos x - cos^3 x = cos x - cos^3 x
Cancel out the common factor cos^2 x:
cos^2 x - sin x cos x = 1 - cos^2 x
Rearrange:
2cos^2 x - sin x cos x - 1 = 0
Unfortunately, this equation does not have a straightforward solution. Further simplification or a numerical method, such as graphing or using a calculator, may be necessary.
21. Sin^2 x - 6sin x + 9 / sin^2 x - 9 = sin x - 3 / sin x + 3:
Combine like terms by finding a common denominator on both sides of the equation:
(sin^2 x - 6sin x + 9)(sin x + 3) = (sin x - 3)(sin^2 x - 9)
Expand and simplify:
sin^3 x + 3sin^2 x - 6sin^2 x - 18sin x + 9sin x + 27 = sin^3 x - 9sin x - 3sin^2 x + 27
Combine like terms:
sin^3 x - 3sin^2 x - 9sin x + 27 = sin^3 x - 9sin x - 3sin^2 x + 27
Rearrange:
0 = 0
This equation holds true for all values of x.