the identity tan^2Φ cos^2Φequals a). minus half b). plus one c). coT^2Φ d). tangentΦ e). all of the above?
To find the identity of tan^2Φ cos^2Φ, we can start by using Pythagorean identity for tangent:
tan^2Φ = sec^2Φ - 1
cos^2Φ = 1 - sin^2Φ
Now, substitute these values into the expression:
tan^2Φ cos^2Φ = (sec^2Φ - 1)(1 - sin^2Φ)
Let's simplify this further:
tan^2Φ cos^2Φ = sec^2Φ - sec^2Φ * sin^2Φ - 1 + sin^2Φ
Notice that sec^2Φ * sin^2Φ can be simplified using another identity:
sec^2Φ * sin^2Φ = 1 - cos^2Φ
Now, let's substitute this value:
tan^2Φ cos^2Φ = sec^2Φ - (1 - cos^2Φ) - 1 + sin^2Φ
Expanding further:
tan^2Φ cos^2Φ = sec^2Φ - 1 + cos^2Φ - 1 + sin^2Φ
Using the fundamental trigonometric identity:
sin^2Φ + cos^2Φ = 1
tan^2Φ cos^2Φ = sec^2Φ - 2
Now, we know that sec^2Φ = 1 + tan^2Φ, so let's substitute this value:
tan^2Φ cos^2Φ = 1 + tan^2Φ - 2
Simplifying further:
tan^2Φ cos^2Φ = tan^2Φ - 1
Therefore, the identity tan^2Φ cos^2Φ equals: tan^2Φ - 1.
So, the correct answer is e). None of the above.