(Sin^3x-cos^3x)/(sinx-cosx) – cosx/sqrt(1+cot^2x)-2tanxcotx=-1 where x∈(0,2pi)
general value of x.
To solve the given equation step-by-step, let's simplify the expression first:
Step 1: Simplify the numerator of the first fraction.
(sin^3x - cos^3x) = (sinx - cosx)(sin^2x + sinxcosx + cos^2x)
= (sinx - cosx)(1 + sinxcosx)
Step 2: Simplify the denominator of the first fraction.
(sinx - cosx)
Step 3: Simplify the second fraction.
cosx/sqrt(1+cot^2x)
Using the identity cot^2x = (cosx/sinx)^2:
= cosx/sqrt(1+(cosx/sinx)^2)
= cosx/sqrt((sin^2x + cos^2x)/sin^2x)
= cosx/(sqrt(sin^2x + cos^2x)/sinx)
= cosx/sinx
Step 4: Simplify the third fraction.
2tanxcotx = 2(sinxcosx)/(cos^2x-sin^2x)
Using the identity cos^2x - sin^2x = cos2x:
= 2sinxcosx/cos2x
= 2sinxcosx/(2cos^2x-1)
Now, let's substitute the simplified expressions into the equation:
[(sinx - cosx)(1 + sinxcosx)]/(sinx - cosx) - cosx/sinx - 2sinxcosx/(2cos^2x-1) = -1
Step 5: Cancel out common factors in the numerator and denominator of the first fraction.
1 + sinxcosx - cosx/sinx - 2sinxcosx/(2cos^2x-1) = -1
Step 6: Get rid of the denominators by multiplying through by sinx.
sinx(1 + sinxcosx) - cosx - 2sin^2xcosx/(2cos^2x - 1) = -sinx
Step 7: Simplify the equation further.
sinx + sin^2xcosx - cosx - 2sin^2xcosx/(2cos^2x - 1) = -sinx
Step 8: Combine like terms.
2sin^2xcosx - cosx/(2cos^2x - 1) = -2sinx
Step 9: Multiply through by (2cos^2x - 1) to get rid of the denominator.
2sin^2xcosx - cosx = -2sinx(2cos^2x - 1)
Step 10: Distribute and simplify.
2sin^2xcosx - cosx = -4sinxcos^2x + 2sinx
Step 11: Simplify further.
2sin^2xcosx + 4sinxcos^2x = cosx + 2sinx
Step 12: Factor out sinx from the left side.
sinx(2sinxcosx + 4cos^2x) = cosx + 2sinx
Step 13: Divide both sides by sinx.
2sinxcosx + 4cos^2x = 1 + 2cosx
Step 14: Rearrange the equation.
4cos^2x - 2cosx + 1 = 0
Step 15: Solve the quadratic equation.
Let's solve the quadratic equation using the quadratic formula:
cosx = (-b ± sqrt(b^2 - 4ac))/(2a)
a = 4, b = -2, c = 1
cosx = (-(-2) ± sqrt((-2)^2 - 4(4)(1)))/(2(4))
cosx = (2 ± sqrt(4 - 16))/8
cosx = (2 ± sqrt(-12))/8
Since we can't take the square root of a negative number, there are no real solutions for cosx.
Therefore, the given equation has no solutions in the interval (0, 2pi).
To find the general values of x that satisfy the equation:
First, let's simplify the equation step by step:
1. Simplify the numerator:
(sin^3x - cos^3x) = (sinx - cosx)(sin^2x + sinxcosx + cos^2x)
2. Simplify the denominator:
sqrt(1 + cot^2x) = sqrt(1 + (cos^2x/sin^2x)) = sqrt((sin^2x + cos^2x)/sin^2x) = 1/sinx
Therefore, the expression becomes:
(sinx - cosx)(sin^2x + sinxcosx + cos^2x) - (cosx/sinx) - 2tanxcotx = -1
3. Expand and simplify:
sin^3x + sin^2xcosx + sinxcos^2x - cosxsin^2x - cos^2xsinx - cosx - 2sinxcosx = -1
Combine like terms:
sin^3x + sin^2xcosx - cos^2xsinx - cosxsin^2x - cos^2x - cosx = -1
4. Rearrange terms and factor out common factors:
sin^2x(sin x + cos x) - cos x (sin^2x + cos x) - (sin^2x + cos x) = -1
Factor out common factors:
(sin^2x - 1)(sin x + cos x) - (sin^2x + cos x) = -1
5. Use the Pythagorean identity sin^2x + cos^2x = 1:
(1 - cos^2x)(sin x + cos x) - (sin^2x + cos x) = -1
Replace (1 - cos^2x) with sin^2x:
sin^2x(sin x + cos x) - (sin^2x + cos x) = -1
6. Rearrange terms:
sin^3x + sin^2x*cosx - sin^2x - cosx - 1 = -1
Combine like terms:
sin^3x + sin^2x*cosx - sin^2x - cosx = 0
7. Factor:
sin^2x (sin x + cos x) - 1(cos x + sin x) = 0
Factor out (sin x + cos x):
(sin x + cos x)(sin^2x - cos x - 1) = 0
Now we have two cases:
Case 1: sin x + cos x = 0
In this case, sin x = -cos x
To find the values of x, we can use the trigonometric ratios. Since x ∈ (0, 2π), we can conclude that:
sin θ = -cos θ, where 0 < θ < 2π.
Solving this equation, we find that:
θ = (3π/4) or θ = (7π/4)
Therefore, the values of x in this case are:
x = (3π/4) and x = (7π/4)
Case 2: sin^2x - cos x - 1 = 0
In this case, we need to solve the quadratic equation for sin x.
Using the quadratic formula, sin x = (-b ± sqrt(b^2 - 4ac))/2a, where:
a = 1, b = -1, c = -1
Substituting the values, we have:
sin x = (-(-1) ± sqrt((-1)^2 - 4(1)(-1)))/2(1)
sin x = (1 ± sqrt(5))/2
Since x ∈ (0, 2π), we need to find the angles within this range that satisfy the above equation:
x = π/10, 3π/10, 9π/10, 11π/10, 17π/10, 19π/10
Therefore, the general values of x that satisfy the given equation are:
x = (3π/4), (7π/4), (π/10), (3π/10), (9π/10), (11π/10), (17π/10), (19π/10).
sin^3 - cos^3 = (sin-cos)(sin^2 + sin*cos + cos^2) = (sin-cos)(1+sin*cos)
so, (sin^3-cos^3)/(sin-cos) = 1+sin*cos
1+cot^2 = csc^2, so
cos/csc = sin*cos
tan*cot=1, so we have
1+sin*cos - sin*cos - 2 = -1
which is true for all values of x