Express f(x) in terms of the function sin(x) and its powers?

Question

f(x) = 6cot^2(x) / cos^2(x) - tan^2(x) / 7sec^2(x)

Can you solve this and show me the most straightforward way of doing so.

Thanks a lot

Answer 1

Make some substitutions using the following identities:
cot x = cos x / sin x
tan x = sin x / cos x
sec x = 1/cos x

So f(x) = 6(cos^2 x / sin^2 x) / cos^2 x - (sin^2 x / cos^2 x) / (7 / cos^2 x)
=6 / sin^2 x - sin^2 x / 7

Answer 2

f(x) = (6cot(x)^2 / cos(x)^2) - (tan(x)^2 / 7sec(x)^2)
f(x) = (6/(cos(x)^2 * tan(x)^2)) - (tan(x)^2 / 7(1/cos(x)^2))
f(x) = (6/(cos(x)^2 * (sin(x)/cos(x))^2)) - ((tan(x)^2)/1)/(7/cos(x)^2))
f(x) = (6/((sin(x)cos(x))^2)/cos(x)^2))) - ((tan(x)^2)/1)*(cos(x)^2 / 7)
f(x) = (6/(sin(x)^2)) - ((tan(x)^2 * cos(x)^2)/7)
f(x) = (6/(sin(x)^2)) - (((sin(x)/cos(x))^2 * cos(x)^2)/7)
f(x) = (6/(sin(x)^2)) - (((sin(x)^2 * cos(x)^2))/(cos(x)^2))/7)
f(x) = (6/(sin(x)^2)) - ((sin(x)^2)/7)

Multiply everything by 7sin(x)^2

f(x) = (42 - sin(x)^4)/(7sin(x)^2)

Answer 3

i assume you advise (sin x + cos x)/cos x ? wherein case you have achieved it incorrect so far, you mustn't have a cos x left. chop up it into (sin x/cos x) + (cos x)/(cos x). you will be waiting to do something.

Answer 4

f(x)=6/sin^2x-sin^2x/7