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sec^6x-tan^6x-3sec^2x*tan^2x=1 how to prove

At 1st I will used the way I know clearly.

Note that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x).

With this, we can transform the left side of this equation to
1/cos^6(x) - sin^6(x)/cos^6(x) - [3/cos^2(x)][sin^2(x)/cos^2(x)].

To put all terms over cos^6(x), we just need to multiply the last term by cos^(x)/cos^2(x).
That gives 1/cos^6(x) - sin^6(x)/cos^6(x) - 3*sin^2(x)cos^2(x)/cos^6(x).

Combining the fractions gives [1 - sin^6(x) - 3*sin^2(x)cos^2(x)]/cos^6(x).

Now the cos^2(x) in the denominator can be changed to 1 - sin^2(x).  This gives
[1 - sin^6(x) - 3*sin^2(x){1 - sin^2(x)}]/cos^6(x).

Multiplying out that 3rd term gives
[1 - sin^6(x) - 3*sin^2(x) + 3sin^4(x)]/cos^6(x).

Reorganizing the numerator gives
[1 - 3*sin^2(x) + 3sin^4(x) - sin^6(x)]/cos^6(x).

The top looks like 1 - 3b + 3b^2 - b^3 where b = sin^2(x).
The expression just given factors into (1-b)^3.

Since b = sin^2(x), this becomes  [1 - sin^2(x)]^3.

It is known that sin^2(x) + cos^2(x) = 1, so subtracting sin^2(x) from both sides gives
cos^2(x) = 1 - sin^2(x).  Noting that this is the same as what's inside the brackets,
[1 - sin^2(x)]^3 become [cos^2(x)]^3.  That is the same as cos^6(x).

Now we get cos^6(x) in the numerator, and that is the same thing as is in the denominator,
so they both cancel and the result is 1.

Now for the 2nd, this could have been done using 1 + tan^2(x) = sec^2(x).

To do it, just put in 1 + tan^2(x) for sec^2(x).
This gives (1 + tan^2)^3 - tan^6(x) - 3(1 + tan^2(x))tan^2(x).

Since (1+a)^3 = 1 + 3a + 3a^2 + a^3, this can be used on the 1st term where a is tan^2(x).
That is, 1 + 3tan^2(x) + 3tan^4(x) + tan^6(x)  -  tan^6(x) - 3tan^2(x) - 3tan^4(x).

Looking at the whole expression, the 2nd term is the negative of the 7th term, the 3rd term is the negative of the 6th term, and the 4th term is the negative of the 5th term.  Cancelling term 2, 7, 3, 7, 4, and 5 leaves us with term 1, and the value of term 1 is 1.

As can be seen, this is easier, but it requires more to be known about the trig functions.

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Scott A Wilson


I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?


Experience in the area; I have tutored people in the above areas of mathematics for over two years in I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.

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