help ASAP !! Precal

Question

pls solve with steps

tan^3 x sec^3 x = (sec^4 x - sec^2 x) sec x tan x

Answer 1

it's often best to convert all to sines & cosines, then simplify

tan^3(x)sec^3(x) = left side

= sin^3(x)/cos^3(x))(1/cos^3(x)

= sin^3(x)

right side has tanxsecx which cancels out Left side leaving tan^2(x)sec^2(x) = sin^2(x)

so, that leaves

sin^2(x) = sec^4(x) -sec^2(x) = sec^2(x)(sec^2(x) -1)

sin^2(x) = (1/cos^2(x))(sec^2(x) -1)

sin^2(x)cos^2(x) = sec^2(x) -1= tan^2(x) = sin^2(x)/cos^2(x)

sin^2(x)cos^2(x) = sin^2(x)/cos^2(x)

divide by sin^2(x) leaving

cos^2(x) = cos^2(x)

QED

Answer 2

Hello Jackie,

When doing these problems where you’re verifying trig identities, I usually start with the side where there’s the most activity. In addition when there is exponents with powersof two or more, the trig identities are gonna come in to play. The main identities are below but keep in mind there are more. You should know these off the top of your head eventually.

1 + (tan x)^2 = (sec x)^2

(sin x)^2 + (cos x)^2 = 1

1 + (cot x)^2 = (csc x)^2

Factor out a (sec x)^2 on the right side

(tan x)^3(sec x)^3 = (sec x)^2 times ((sec x)^2 - 1)secxtanx

now divide both sides by secxtanx and get

(tan x)^2 times (sec x)^2 = (sec x)^2 times ((sec x)^2 -1)

substitute (tan x)^2 for (sec x)^2 -1

And the simplifies to

((tan x)^2)((sec x)^2 = (sec x)^2 (tan x)^2

Proven

Answer 3

Answer 4

Recall that 1 + tan²x = sec²x.

So, (sec⁴x - sec²x)secx tanx = sec²x(sec²x - 1)secx tanx = sec²x tan²x secx tanx = sec³xtan³x