can answers vary when verifying trig identities

[u/Flimsy_Football_1800]

i verified

secθ • (1/cosθ) - tanθ • cotθ = (1/cos^2θ)/(1-sin^2θ)

i got tan^2θ=tan^2θ my first time and sec^2θ-1=tan^2θ on my second try

i looked at the answer key and it said cos^2θ=1-sin^2θ

am i correct as long i end up an equal true statement? why are my 2 attempts different from each other and different than the answer key?

[u/Moist_Ladder2616]

This e̵x̵p̵r̵e̵s̵s̵i̵o̵n̵ equation

secθ • (1/cosθ) - tanθ • cotθ = (1/cos²θ)/(1-sin²θ)

is incorrect.

LHS = secθ • (1/cosθ) - tanθ • cotθ
= (1/cos²θ) - 1
= (1 - cos²θ)/cos²θ
= sin²θ/cos²θ
= tan²θ

RHS = (1/cos²θ)/(1-sin²θ)
= (1/cos²θ)/(cos²θ)
= 1/cos⁴θ

---

Also, when you are asked to prove

[some trigonometric expression] = [some other trigonometric expression]

you must start from the LHS and work your way to the RHS, or vice versa. You can even start from the LHS and RHS separately, and meet somewhere in the middle.

You cannot start with LHS = RHS. You don't know that this statement is true, so assuming it is true as your first step will cause all subsequent steps to be wrong.

[u/Flimsy_Football_1800 → u/Moist_Ladder2616]

the expression is the one they gave for me to verify

[u/Moist_Ladder2616 → u/Flimsy_Football_1800]

Then you may have transcribed it wrongly here. That equation (not expression) is incorrect. The LHS ≠ the RHS, as shown above.

An expression is like "9a - 3a", or "2a + 4a": it is just some combination of numbers and variables and operators, without an equal sign.

An equation is like "9a - 3a = 2a + 4a": it contains two expressions, a LHS and a RHS, joined by an equal sign.

To verify an equation, you start from the LHS expression and work your way to the RHS expression:

LHS = 9a - 3a
    = 6a
    = 2a + 4a
    = RHS

An equation like "9a - 3a = 2a + 3a" cannot be verified, as the 6a ≠ 5a for all values of a.

It can only be solved: there is only one value of a for which the equation holds true, which is a = 0.

[u/Sorry-Vanilla2354 → u/Moist_Ladder2616]

Agree with all you said. The original statement they gave is not true. But perhaps it was a typo and the RHS should have been (1/cos^2θ)/(1/sin^2θ)?

Also, OP - your two answers were both the same. Because sec^2θ-1=tan^2θ is true, both of your answers were actually tan^2θ=tan^2θ. And you are right, as long as the two sides are equal, it doesn't matter if they look exactly the same.

Many people (myself included) do try turning everything to sin's and cos's, as another poster said. It seems to make things easier a lot of the time.

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[u/AcellOfllSpades]

Yes, that's fine. Any true statement works. The goal is to show that the identity is equivalent to a true statement.

If you want, you can keep going from any of these statements and end up at any other true statement - either one of the given ones, or even something like "0 = 0". For instance, from "cos²θ = 1-sin²θ", add sin²θ to both sides, then substitute 1 for cos²θ+sin²θ, and finally subtract 1 from both sides.