How do you verify the equation is an identity?
Tan^2x-tan^2y=sec^2x-sec^2y
and, how do you factor and simplify,
cscx(sin^2x+cos^2xtanx)/sinx+cosx
One of the basic identities is
tan^2 A + 1 = sec^2 A which gives Tan^2 A = sec^2 A - 1
so
Left side
= tan^2x-tan^2y
= sec^2x - 1 -(sec^2y - 1)
= sec^2x - sec^2y
= Right Side
To verify if the equation is an identity, you can start by simplifying both sides of the equation and checking if they match.
1. Verification of the Equation:
Start with the left side: Tan^2x - Tan^2y
By a trigonometric identity, we know that Tan^2x = sec^2x - 1 and Tan^2y = sec^2y - 1.
Substitute these results into the equation:
(sec^2x - 1) - (sec^2y - 1) = sec^2x - sec^2y
Simplify both sides:
sec^2x - 1 - sec^2y + 1 = sec^2x - sec^2y
sec^2x - sec^2y = sec^2x - sec^2y
Since both sides of the equation are equal, the equation is verified as an identity.
2. Factoring and Simplifying:
To factor and simplify the expression cscx(sin^2x + cos^2xtanx)/(sinx + cosx), we can follow these steps:
First, notice that sin^2x + cos^2x equals 1 due to the Pythagorean identity.
The expression can now be written as: cscx(1 + cos^2xtanx)/(sinx + cosx)
Now, let's factor out cosx from the numerator and sinx from the denominator:
cscx(cos^2x + 1 * tanx)/(sinx * (1 + cotx))
Using the reciprocal identity: cotx = cosx/sinx, we can simplify further:
cscx(cos^2x + tanx)/(sinx + cosx)
This is the simplified form of the expression.
To verify whether the equation is an identity, you need to show that the equation holds true for all values of x and y.
For the equation Tan^2x - tan^2y = sec^2x - sec^2y, you can start by simplifying both sides of the equation using the trigonometric identities:
1. Recall the Pythagorean identity: sin^2x + cos^2x = 1.
2. Divide the equation by cos^2x to obtain: tan^2x + 1 = sec^2x.
3. Repeat the same steps for the term involving y to have: tan^2y + 1 = sec^2y.
Now, substitute these simplifications back into the original equation:
tan^2x - tan^2y = sec^2x - sec^2y
(tan^2x + 1) - (tan^2y + 1) = (sec^2x) - (sec^2y)
(sec^2x) - (sec^2y) = (sec^2x) - (sec^2y)
Since the left-hand side is equal to the right-hand side, the equation holds true for all values of x and y. Therefore, it is an identity.
Regarding the factoring and simplification of the expression:
cscx(sin^2x + cos^2x*tanx) / (sinx + cosx)
1. Simplify sin^2x + cos^2x*tanx:
Since cos^2x = 1 - sin^2x (by the Pythagorean identity), replace cos^2x in the expression:
sin^2x + (1 - sin^2x)*tanx = sin^2x + tanx - sin^2x*tanx.
2. Simplify the denominator, (sinx + cosx), by using the reciprocal identity:
(1/sinx)*(sinx + cosx) = (1 + cosx/sinx).
3. Now, substitute the simplified numerator and denominator back into the expression:
[ cscx*(sin^2x + tanx - sin^2x*tanx) ] / (1 + cosx/sinx)
4. Further simplification can be done by distributing cscx into the numerator:
[ cscx*sin^2x + cscx*tanx - cscx*sin^2x*tanx ] / (1 + cosx/sinx)
5. Next, combine the like terms cscx*sin^2x and -cscx*sin^2x*tanx:
[ cscx*sin^2x*(1 - tanx) + cscx*tanx ] / (1 + cosx/sinx)
6. Finally, factor out cscx from the numerator:
[ cscx*(sin^2x*(1 - tanx) + tanx) ] / (1 + cosx/sinx)
This is the factored and simplified form of the expression cscx*(sin^2x + cos^2xtanx) / (sinx + cosx).
For the second, I will assume you meant
cscx(sin^2x+cos^2xtanx)/(sinx+cosx)
= 1/sinx(sin^2x + cos^2xsinx/cosx)/(sinx + cosx)
= 1/sinx(sin^2x + sinxcosx)/(sinx+cosx)
= (sinx + cosx)/sinx + cosx)
= 1