Please Help!

[Novice]

If 10 sin⁴ x + 15 cos⁴ x = 6

find the value of

27 cosec⁶ x + 8 sec⁶ x


I am trying to reduce the latter to some form where I can substitute the value of the former equation and get the answer, however I have failed continuously in my attempts. Please Help!

 

[biffboy]

If 10 sin⁴ x + 15 cos⁴ x = 6

find the value of

27 cosec⁶ x + 8 sec⁶ x


I am trying to reduce the latter to some form where I can substitute the value of the former equation and get the answer, however I have failed continuously in my attempts. Please Help!

cos^4x=(1-sin^2x)^2=1-2sin^2x+sin^4x Substitute this for cos^4x in the first line.

 

[Deleted member 4993]

If 10 sin⁴ x + 15 cos⁴ x = 6

find the value of

27 cosec⁶ x + 8 sec⁶ x


I am trying to reduce the latter to some form where I can substitute the value of the former equation and get the answer, however I have failed continuously in my attempts. Please Help!

Looks like you want to get to form where:

27cosec⁶x + 8sec⁶x = N * (10sin⁴x + 15cos⁴x)

and you want to find the value of N → find LHS.

That could be a tall order.

If you are not bound to follow that process, follow biffboy's method and:

transform the first equation into a quadratic of sin²x → solve for sin²x and cos²x → substitute into 2nd equation and simplify.

 

[Novice]

Thank You for your suggestions. I sure get the answer that way, although I was hoping there was some beautiful solution to the same. ( I think that is because I am accustomed to tackling difficult looking problems with very easy and elegant solutions). Thanks anyway!

 

[mmm4444bot]

I was hoping there was some beautiful solution

I am accustomed to tackling difficult looking problems with very easy and elegant solutions

If you were to say these things in your original post, then we would know what you were looking for.

In the absence of any questions asked or reasoning offered, it is difficult for volunteers on these boards to determine a student's situation, need, or expectation.

Please consider this for next time. Cheers ~ Mark :cool:

 

[Deleted member 4993]

possible path to elegant answer:

3³= 27

2³ = 8

a³ + b³ = (a + b)(a² - ab + b²)

 

[Novice]

If you were to say these things in your original post, then we would know what you were looking for.

In the absence of any questions asked or reasoning offered, it is difficult for volunteers on these boards to determine a student's situation, need, or expectation.

Please consider this for next time. Cheers ~ Mark :cool:

"I am trying to reduce the latter to some form where I can substitute the value of the former equation and get the answer."
I had written this in my original post! Cheers!

 

[Novice]

possible path to elegant answer:

3³= 27

2³ = 8

a³ + b³ = (a + b)(a² - ab + b²)

Already tried that. Doesn't help. I have come to the conclusion that there is no "elegant" solution to the problem. The one suggested by biffboy or others of the like are the only ways of arriving at the solution.
Thanks for your help!

 

[mmm4444bot]

I had written this

Your "this" is not my point. I quoted the statements to which I referred.

Here's my point, basically: when posters include something about their motivation or expectations up-front, they will receive more helpful responses sooner and volunteers may avoid typing stuff that the poster has already learned or tried.

Cheers!! :cool:

 

[Novice]

Your "this" is not my point. I quoted the statements to which I referred.

Here's my point, basically: when posters include something about their motivation or expectations up-front, they will receive more helpful responses sooner and volunteers may avoid typing stuff that the poster has already learned or tried.

Cheers!! :cool:

Cheers!