1. (sinx/cscx)+(cosx/secx)=1
2. (1/sinxcosx)-(cosx/sinx)=tanx
3. (1/1+cos s)=csc^2 s-csc s cot s
4. (secx/secx-tanx)=sec^2x+secxtanx
5. (cosx/secx-1)-(cosx/tan^2x)=cot^2x
sinx/cscx is the same thing as sin^2 x
cosx/secx is the same thing as cos^2 x
What you you know about the sum of sin^2 and cos^2 ?
Try proving the other identities yourself, writing tan, csc and sec in terms of sin and cos.
To solve each of these equations, we'll start by simplifying both sides of the equation and identify any identities or properties that can be applied. Then, we'll manipulate the equation to isolate the variable, x.
1. (sinx/cscx) + (cosx/secx) = 1
First, simplify the left side:
sinx/cscx = sinx/sinx = 1
cosx/secx = cosx/cosx = 1
Now, substitute these simplified forms back into the original equation:
1 + 1 = 1
So, the left side does equal the right side. Therefore, the equation is true for all x.
2. (1/sinxcosx) - (cosx/sinx) = tanx
First, find a common denominator for the terms on the left side:
(1/sinxcosx) - (cosx/sinx) = (sinx - cos^2x) / (sinxcosx)
Now, simplify the numerator using the Pythagorean identity sin^2x + cos^2x = 1:
(sin^2x - cos^2x) / (sinxcosx)
Again, using the Pythagorean identity sin^2x = 1 - cos^2x:
((1 - cos^2x) - cos^2x) / (sinxcosx)
Simplify further:
(1 - 2cos^2x) / (sinxcosx)
Now, use the identity 1 - 2sin^2x = -cos(2x):
(-cos2x) / (sinxcosx)
Use the identity sin2x = 2sinxcosx:
-1 / 2tanx
Therefore, the equation simplifies to -1 / 2tanx = tanx.
To isolate tanx, we can multiply both sides by 2tanx:
-1 = 2tan^2x
Finally, divide both sides by 2 to solve for tanx:
tan^2x = -1/2
tanx = ±√(-1/2)
So, the equation has no real solutions for x.
3. (1/(1 + cosx)) = csc^2x - cscx cotx
In this equation, start by simplifying the right side using various trigonometric identities.
Recall that cscx = 1/sinx, cotx = cosx/sinx, and csc^2x = (1/sinx)^2 = 1/sin^2x = cosec^2x:
csc^2x - cscx cotx = (1/sin^2x) - (1/sinx)*(cosx/sinx)
= 1/sin^2x - cosx/sin^2x
= (1 - cosx)/sin^2x
Now, substitute this simplified form back into the original equation:
(1/(1 + cosx)) = (1 - cosx)/sin^2x
To simplify this equation further, invert both sides:
(1 + cosx) = sin^2x/(1 - cosx)
Now, cross-multiply to eliminate the fractions:
(1 + cosx)(1 - cosx) = sin^2x
Simplify the left side:
1 - cos^2x = sin^2x
Using the Pythagorean identity cos^2x + sin^2x = 1:
1 - 1 + sin^2x = sin^2x
This simplifies to 0 = 0, which means that the left side equals the right side. Therefore, the equation is true for all x.
4. (secx/(secx - tanx)) = sec^2x + secx tanx
First, simplify the left side by multiplying both the numerator and denominator by secx - tanx:
secx * [(secx - tanx)/(secx - tanx)] = sec^2x - sectanx
Now, substitute this simplified form back into the original equation:
sec^2x - sectanx = sec^2x + secx tanx
To simplify further, cancel out the sec^2x terms on both sides of the equation:
- sectanx = secx tanx
Divide both sides of the equation by -tanx to isolate secx:
secx = -1
Since secx equals -1, this means that the equation has no real solutions for x.
5. (cosx/(secx - 1)) - (cosx/(tan^2x)) = cot^2x
First, simplify the left side by finding a common denominator for the two terms:
cosx * (tan^2x) - cosx * (secx - 1) / (secx - 1) * (tan^2x)
Multiply out the numerator:
cosx * tan^2x - cosx * secx + cosx / (secx - 1) * (tan^2x)
Now, simplify further using various trigonometric identities:
tan^2x = (sinx/cosx)^2 = sin^2x/cos^2x
secx = 1/cosx
Substitute these identities back into the equation:
cosx * sin^2x/cos^2x - cosx * (1/cosx) + cosx / (1 - cosx) * (sin^2x/cos^2x)
Simplify the numerator:
cosx * sin^2x - cosx + cosx * sin^2x / cos^2x / (1 - cosx)
Combine the two fractions:
(cosx * sin^2x - cosx * cos^2x + cosx * sin^2x) / (cos^2x * (1 - cosx))
Simplify the numerator:
(cosx * sin^2x - cos^3x + cosx * sin^2x) / (cos^2x * (1 - cosx))
Combine like terms:
2 cosx * sin^2x - cos^3x / (cos^2x * (1 - cosx))
Use the identity sin^2x = 1 - cos^2x:
2 cosx (1 - cos^2x) - cos^3x / (cos^2x * (1 - cosx))
Simplify further:
2 cosx - 2cos^3x - cos^3x / (cos^2x * (1 - cosx))
Combine the similar terms in the numerator:
2 cosx - 3cos^3x / (cos^2x * (1 - cosx))
Finally, simplify further:
2 - 3cos^2x / (1 - cosx)
Now, the equation simplifies to cot^2x = 2 - 3cos^2x / (1 - cosx).