Show what happens to 'U' if 'a' is doubled.

[SalfordPhysics]

Homework Statement

Demonstrate what happens to X when a is doubled (a → 2a).

a = tan^2(X) - 1/cos^2(X)

Homework Equations

None given formally;
Pythagorean Theorem - sin^2(θ) + cos^2(θ) = 1 (added myself).

The Attempt at a Solution

Divide through Pythagorean above to give: tan^2(θ) + 1 = 1/cos^2(θ)
1/cos^2(θ) = sec^2(θ).
tan^2(θ) - sec^2(θ) = -1

From given equation, a = -1.

Doubling a yields 2a.
But tan^2(θ) - sec^2(θ) = -1 for all values of (θ) including X as given.

Is my solution therefore that nothing happens to X if a is doubled, or does the solution not consist of using the pythagorean identity?

All thanks appreciated.

 

[Dick]

Homework Statement

Demonstrate what happens to X when a is doubled (a → 2a).

a = tan^2(X) - 1/cos^2(X)

Homework Equations

None given formally;
Pythagorean Theorem - sin^2(θ) + cos^2(θ) = 1 (added myself).

The Attempt at a Solution

Divide through Pythagorean above to give: tan^2(θ) + 1 = 1/cos^2(θ)
1/cos^2(θ) = sec^2(θ).
tan^2(θ) - sec^2(θ) = -1

From given equation, a = -1.

Doubling a yields 2a.
But tan^2(θ) - sec^2(θ) = -1 for all values of (θ) including X as given.

Is my solution therefore that nothing happens to X if a is doubled, or does the solution not consist of using the pythagorean identity?

All thanks appreciated.

You have it right. a=(-1) is the only solution. You can't really even talk about what happens to X when a=(-2), then there are no solutions for X.

 

[Mark44]

Homework Statement

Demonstrate what happens to X when a is doubled (a → 2a).

a = tan^2(X) - 1/cos^2(X)

Homework Equations

None given formally;
Pythagorean Theorem - sin^2(θ) + cos^2(θ) = 1 (added myself).

The Attempt at a Solution

Divide through Pythagorean above to give: tan^2(θ) + 1 = 1/cos^2(θ)
1/cos^2(θ) = sec^2(θ).
tan^2(θ) - sec^2(θ) = -1

From given equation, a = -1.

Doubling a yields 2a.
But tan^2(θ) - sec^2(θ) = -1 for all values of (θ) including X as given.

Is my solution therefore that nothing happens to X if a is doubled, or does the solution not consist of using the pythagorean identity?

All thanks appreciated.

Looks fine, if a bit long-winded.
You're given a = tan²(x) - 1/cos²(x)
The right side can be written as tan²(x) - sec²(x) = tan²(x) - (tan²(x) + 1) = tan²(x) - tan²(x) - 1 = -1

So a = -1 identically for all values of x for which tan(x) is defined and cos(x) ≠ 0.

Doubling a has no effect on x.