Simplify the expression (tanθ+cotθ)^2 - (tanθ-cotθ)^2 using algebra and trigonometric identities.
(tanθ+cotθ)^2 - (tanθ-cotθ)^2
= (tan^2θ + 2 + cot^2θ) - (tan^2θ - 2 + cot^2θ)
= 4
because tanθ cotθ = 1
To simplify the given expression, we'll use the algebraic identity:
(a + b)^2 = a^2 + 2ab + b^2
Let's rewrite the given expression using this identity:
(tanθ + cotθ)^2 - (tanθ - cotθ)^2
Using the identity, the expression becomes:
(tanθ)^2 + 2(tanθ)(cotθ) + (cotθ)^2 - [(tanθ)^2 - 2(tanθ)(cotθ) + (cotθ)^2]
Now let's simplify further. Recall the trigonometric identity:
tanθ = sinθ/cosθ,
cotθ = cosθ/sinθ
Replace (tanθ)(cotθ) using these identities:
(tanθ)^2 + 2(tanθ)(cotθ) + (cotθ)^2
= (sinθ/cosθ)^2 + 2(sinθ/cosθ)(cosθ/sinθ) + (cosθ/sinθ)^2
= sin^2θ/cos^2θ + 2cosθsinθ/sinθcosθ + cos^2θ/sin^2θ
Using the trigonometric identity:
sin^2θ + cos^2θ = 1
We can substitute cos^2θ with 1 - sin^2θ:
= sin^2θ/cos^2θ + 2cosθsinθ/sinθcosθ + (1 - sin^2θ)/sin^2θ
Now let's simplify each term:
sin^2θ/cos^2θ = tan^2θ,
2cosθsinθ/sinθcosθ = 2,
(1 - sin^2θ)/sin^2θ = 1/sin^2θ - 1
Now substituting these simplified terms back into the expression:
tan^2θ + 2 - 1/sin^2θ + 1
Combining the terms gives us the simplified expression:
tan^2θ + 1/sin^2θ + 3
So, the simplified expression is tan^2θ + 1/sin^2θ + 3.