(1_ cos ^2)(1_tan^2)=tan^2
Assuming you mean (1 - cos^2 θ)(1 -tan^2 θ) = tan^2 θ:
There are two Pythagorean Identities that can be applied here, 1+cos^2 θ = sin^2 θ, and 1 - tan^2 θ = sec^2 θ.
Thus, we have (sin^2 θ)(sec^2 θ) = tan^2 θ
Since sec^2 θ = 1/cos^2 θ, then we have sin^2 θ/cos^2 θ = tan^2 θ and they are indeed equal to each other by the Quotient Identity.
Correction: 1+tan^2θ = sec^2θ
really? This is just
(sin^2)(sec^2)
need I say more?
and that's -, not _
Assuming you meant:
(1 - cos^2 θ)(1 -tan^2 θ) = tan^2 θ
Secondly, you don't say what to do with this, I will assume
you want to solve, since it is not an identity
Thirdly, you didn't state a domain , so I will assume 0 ≤ x ≤ 2π
1 - tan^2 θ - cos^2 θ + (cos^2 θ)(tan^2 θ) = tan^2 θ
1 - sin^2 θ/cos^2 θ - cos^2 θ + sin^2 θ = sin^2 θ/cos^2 θ
multiply by cos^2 θ
cos^2 θ - sin^2 θ - cos^4 θ + (1 - cos^2 θ)(cos^2 θ) = 1 - cos^2 θ
cos^2 θ - 1 + cos^2 θ - cos^4 θ + cos^2 θ - cos^4 θ = 1 - cos^2 θ
-2cos^4 θ + 4cos^2 θ - 2 = 0
cos^4 θ - 2cos^2 θ + 1 = 0
(cos^2 θ - 1) = 0
cos^2 θ = 1
cosθ = ± 1
θ = 0, π, 2π
pwhewww!
Argghhh,
always trust oobleck to cut through the fog.
But the actual question contained
1 - tan^2 θ which is NOT equal to sec^2 θ
so my solution is right after all,
The 1+tan^2θ was only found in the title, not in the question
To prove the given statement, we need to simplify the expression: (1 - cos^2)(1 - tan^2).
Let's rewrite this expression using trigonometric identities:
(1 - cos^2)(1 - tan^2)
Using the Pythagorean identity for cosine (1 - cos^2) = sin^2:
(sin^2)(1 - tan^2)
Now, applying the Pythagorean identity for tangent (tan^2 + 1 = sec^2):
(sin^2)(sec^2 - 1)
Next, we can simplify the expression further:
(sin^2)(sec^2 - 1)
(sin^2)(sec^2) - (sin^2)
sin^2 * sec^2 - sin^2
Now, let's use another trigonometric identity: (sin^2 + cos^2 = 1) and rearrange it to get (1 - cos^2 = sin^2). So we can rewrite the expression:
sin^2 * sec^2 - sin^2
Replace sec^2 with (1 + tan^2), using the Pythagorean identity for secant:
sin^2 * (1 + tan^2) - sin^2
Expanding the expression:
sin^2 + sin^2 * tan^2 - sin^2
The sin^2 terms cancel out:
0 + sin^2 * tan^2 - 0
sin^2 * tan^2
Finally, we end up with:
tan^2
Thus, we have proved that (1 - cos^2)(1 - tan^2) simplifies to tan^2.