Is sec x^2 equal to (sec x)^2?

Question

If so, the more information why, the better. If not, please explain why. Thanks in advance.

Answer 1

They are not necessarily equal... (They might be equal for specific values of x, but they aren't equal all the time.)

In sec x^2, the exponent is on the x.
In (sec x)^2, the exponent is on the function (sec).

As an example... Let x=20

sec x^2 = sec (20)^2 = sec 400 = 1.3054...
(sec x)^2 = (sec 20)^2 = (1.06418...)^2 = 1.13247...

Answer 2

No, not necessarily. Consider x = π. sec²(π) = (-1)² = 1. But
sec(π²) doesn't equal 1, because the only values of x where sec(x) = 1 is when x = 0, 2π, 4π, 6π, etc., and π² doesn't fall into that series. Sec(π²) is actually about -1.107. On the other hand, when x=0, sec(0) = 1, so sec(0²) = sec²(0).

Answer 3

NO.

sec^2 x = (sec x)^2, but
neither is = to sec x^2.

Answer 4

no
sec x^2 means take the sec of (x^2)
and (sec x) ^2 means take the sec of x and square the result
for instance sec pi is -1 so (sec pi)^2 =(-1)^2 = 1
but the sec of (pi^2) is sec 9.8596=-1.107805712

another example
(sec (pi/2)) ^2 is undefined
but sec (pi/2)^2=-1.280062439

Answer 5

No. Not as functions, anyway; obviously they coincide at x=0, for instance.

I don't think there's really much need for explanation. This shouldn't be a surprise; there are very few functions f(x) for which f(x^2) = (f(x))^2. Off-hand, all I can think of are f(x) = 0, f(x) = 1 and f(x) = x, though there are probably others.

Answer 6

No they are not equal. Let's use cosine for example because it easily works in a calculator.

Let's say x=3

cos3^2= cos9= 0.9876...

(cos3)^2= (0.9986...)^2= 0.9972...

This is proven for secant too because secant is a reciprocal of cosine.