Why is cos(sin(tan(log(lnx))))=1 for all values of x?

let ln x = a

log(ln x) = b or log a = b

tan(log(ln x)) = c or tan(b) = c

cos(tan(log(ln x))) = d or cos(c) = d

Then, sin d = 0.0174 so d = 0.01740087...

Now, cos(c) = d so c =1.553394...

So, tan(b) = c or b= 0.99882....

Now, log a = b so a = 9.9730....

So, ln x = a or x = 21439.95111

So this is the answer x = 21439.95111

So this is the only value(approximate) where the function equals this number. Now, you may find a lot of values close to this, but they are not the same exact number, just close to it.

The reason they are close, is that the ln(log) transform very big values to small ones in a very huge way, so that no matter how big a number that you put in this function, you get very small numbers close to zero. Then, for these small values tan will not change appreciably these values, the same for sin and then cos will receive small values which it will transform to almost 1 since cos(0)= 1.

To see the big difference in the values of this function, you need a supermega calculator!

Let

y = log(ln x)

Then ln x = 10^y and x = exp(10^y)

So, unless x is very very large, y is going to be close to zero.

Now, if z is close to zero, we have the following fairly decent approximations for the trig functions (assuming your calculator uses degrees):

tan z ~ pi * z /180

sin z ~ pi * z / 180

cos z ~ 1 - (z*pi/180)^2/2

So the value of

sin (tan y)

is approximately:

y * (pi/180)^2

So cos (sin (tan y)) will be very close to 1 - (y*(pi/180)^3)^2/2

which is:

1 - y^2 * (pi/180)^6 / 2

or about:

1 - y^2/70756410051

So unless y is significantly more than 10, this value is going to be very close to 1. But if y>10, then x = exp(10^y) > exp(10000000000) which is a very large number.

The second part is the same, except this time, you are essentially getting sin(cos(1)).

1) cos(sin(tan(log(lnx)))) = 1

Let A = sin(tan(log(lnx))).

Then, cos(A) = 1.

Therefore, A = 0.

Thus , sin(tan(log(lnx))) = 0

Let B = tan(log(lnx)).

Then, sin(B) = 0.

Therefore, B = 0.

Thus, tan(log(lnx)) = 0

Let C = log(lnx).

Then, tan(C) = 0.

Therefore, C = 0.

Thus, log(lnx) = 0.

Let D = lnx.

Then, log(D) = 0.

Therefore, D = 10^0 = 1

Thus, lnx = 1

So, x = e^1 = e = 2.718281828...

This appears to be the only true value for x.

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2) sin(cos(tan(log(lnx)))) = 0.017452406...

In a similar way to the above,

cos(tan(log(lnx))) = 1

tan(log(lnx)) = 0

log(lnx) = 0

lnx = 10^0 = 1

x = e^1 = e = 2.718281828...

lnx is much smaller than x for x > 1

log y is much smaller than y

So log(lnx) is extremely much smaller than x

tan(almost 0) = almost 0

cos(almost 0) = very near 1

So your statement is not exactly true.

Guess x = e^1000. Then ln(x) = 1000

log1000 = 3

3 is very much more smaller than e^1000

tan(3 rad) = ... From here you see your statement goes wrong.

Th

log and lnx cancel each other out and sin x tan = sin x (sin/cos) which gives 1/cos and then you have the cos on the outside which would give you a cos/cos which of course is equal to 1.

Your function is undefined for x ≤ 1, and does not equal 1 for any x > 1.

Neither is your 2nd function correct

The above is not a true statement. It is, for example, not true for x = -1, for which the above function is undefined.