1. (sec^2x-6tanx+7/sec^2x-5)=(tanx-4/tanx+2)
2. (sin^3A+cos^3A/sinA+cosA)=1-sinAcosA
3. csc^6x-cot^6x+1+3csc^2xcot^2x
please help
Did you leave out an = sign in #3 ?
To simplify these expressions, we will use trigonometric identities. Let's solve them one by one.
1. To simplify the expression (sec^2x - 6tanx + 7) / (sec^2x - 5) = (tanx - 4) / (tanx + 2), we'll start by substituting the values of sec^2x and tanx in terms of sine and cosine:
Recall that sec^2x = 1/cos^2x and tanx = sinx/cosx.
So, the expression becomes:
(1/cos^2x - 6sinx/cosx + 7) / (1/cos^2x - 5) = (sinx/cosx - 4) / (sinx/cosx + 2)
Next, we'll simplify the expression by multiplying both sides by (cos^2x - 5) to get rid of the denominators:
(1 - 6sinx*cosx + 7cos^2x) = (sinx - 4cosx) * (cos^2x + 2)
Simplifying both sides further gives:
1 - 6sinx*cosx + 7cos^2x = sinx*cos^2x + 2sinx - 4cosx - 8cos^2x
Rearranging the terms and grouping similar terms, we get:
8cos^2x + 7cos^2x + 6sinx*cosx + 4cosx - sinx*cos^2x - 2sinx = 1
Now, we use trigonometric identities to further simplify the expression:
Recall that sin^2x + cos^2x = 1 and cos^2x = 1 - sin^2x.
Substituting cos^2x with 1 - sin^2x, we get:
8(1 - sin^2x) + 7(1 - sin^2x) + 6sinx*cosx + 4cosx - sinx(1 - sin^2x) - 2sinx = 1
Simplifying further gives:
8 - 8sin^2x + 7 - 7sin^2x + 6sinx*cosx + 4cosx - sinx + sin^3x - 2sinx = 1
Rearranging and grouping similar terms:
16sin^3x - 15sin^2x + 8sinx*cosx + 3sinx - 2 = 0
This is the simplified form of the expression.
2. To simplify the expression (sin^3A + cos^3A) / (sinA + cosA) = 1 - sinAcosA, we'll use the sum of cubes formula (a^3 + b^3 = (a + b)(a^2 - ab + b^2)).
Using the identity sin^2A + cos^2A = 1, we can rewrite the expression as:
(sin^3A + cos^3A) / (sinA + cosA) = (sinA + cosA)(sin^2A - sinAcosA + cos^2A) / (sinA + cosA)
Notice that the term (sinA + cosA) cancels out in the numerator and the denominator, so the expression becomes:
sin^2A - sinAcosA + cos^2A = 1 - sinAcosA
Since sin^2A + cos^2A = 1, we can simplify it to:
1 - sinAcosA = 1 - sinAcosA
Hence, the expression is simplified to an identity, which means it holds true for all values of A.
3. To simplify the expression csc^6x - cot^6x + 1 + 3csc^2x*cot^2x, we will use trigonometric identities.
Recall that csc^2x = 1/sin^2x and cot^2x = cos^2x/sin^2x.
Substituting these values in the expression gives:
(1/sin^2x)^3 - (cos^2x/sin^2x)^3 + 1 + 3(1/sin^2x)(cos^2x/sin^2x)
Simplifying further:
1/sin^6x - (cos^6x/sin^6x) + 1 + 3cos^2x/sin^4x
Combining like terms:
(1 - cos^6x + sin^6x + 3cos^2x) / sin^6x
Using the identity sin^2x + cos^2x = 1, we can modify the expression:
(1 - cos^6x + (1 - cos^2x)^3 + 3cos^2x) / sin^6x
Simplifying the exponent and expanding:
(1 - cos^6x + (1 - 2cos^2x + cos^4x) + 3cos^2x) / sin^6x
Grouping similar terms:
(2 - cos^6x + cos^4x + 4cos^2x) / sin^6x
We've simplified the expression to its final form.