1. (2cosx-2secx)(3cosx^2+3secx^2+β)
2. (16tanxsecx-4secx)/(8tanxsecx+4secx)
3.sinxcosx^3+cosx^4 (hint: GCF)
4. (cosβ+sinβ)^2
Trigonometric identities involve an equal sign with a different expressions on either side.
So it's not clear what exactly you require in your questions. Do you need to simplify the expressions?
I suggest you check for typos as well as I do not see what β has to do with question 1.
I apologize; it is simplifying. The homework is for my identities unit, though. And number 1 does not have a typo. Sorry!
Typically with simplification like these, you would convert all functions into sine and cosine, and simplify from there on. Use sin²x+cos²x=1 whenever appropriate.
1. To simplify the expression (2cosx-2secx)(3cosx^2+3secx^2+β), we can start by distributing the first term, 2cosx, to each term inside the second parentheses. Similarly, we can distribute the second term, -2secx, to each term inside the second parentheses. This gives us:
(2cosx)(3cosx^2 + 3secx^2 + β) - (2secx)(3cosx^2 + 3secx^2 + β)
Now, we can simplify each term separately:
First term:
(2cosx)(3cosx^2) + (2cosx)(3secx^2) + (2cosx)(β)
= 6cos^3(x) + 6cosxsec^2(x) + 2βcosx
Second term:
-(2secx)(3cosx^2) - (2secx)(3secx^2) - (2secx)(β)
= -6cosx^2secx - 6sec^3(x) - 2βsecx
Finally, we can combine like terms:
6cos^3(x) - 6cosx^2secx + 6cosxsec^2(x) - 6sec^3(x) + 2βcosx - 2βsecx
2. To simplify the expression (16tanxsecx-4secx)/(8tanxsecx+4secx), we can start by factoring out a common factor from both the numerator and denominator.
Numerator:
16tanxsecx - 4secx = 4secx(4tanx - 1)
Denominator:
8tanxsecx + 4secx = 4secx(2tanx + 1)
Now we can cancel out the common factor of 4secx:
(4secx(4tanx - 1))/(4secx(2tanx + 1))
Simplifying further by canceling out the common factor of 4secx:
(4tanx - 1)/(2tanx + 1)
3. To simplify the expression sinxcosx^3 + cosx^4 using the greatest common factor (GCF) method, we need to find the highest power of each term present.
The highest power of sinx is 1, and the highest power of cosx is 4. Therefore, the GCF can be cosx raised to the power of 1.
Next, we divide each term by the GCF:
sinxcosx^3 / cosx^1 + cosx^4 / cosx^1
simplifies to:
sinx cosx^2 + cosx^3
4. To simplify the expression (cosβ+sinβ)^2, we can first expand it using the square of a binomial formula:
(cosβ + sinβ)^2 = (cosβ)^2 + 2(cosβ)(sinβ) + (sinβ)^2
Simplifying each term:
(cosβ)^2 = cos^2(β)
2(cosβ)(sinβ) = 2cos(β)sin(β)
(sinβ)^2 = sin^2(β)
Therefore, the simplified expression becomes:
cos^2(β) + 2cos(β)sin(β) + sin^2(β)