sin2θ = k, then the value of tan^3θ/1 + tan^2θ + cot^3θ/1 + cot^2θ =
Answers
ANSWER:
Given:
- sin2θ = k
To Find:
- Value of tan³θ/1 + tan²θ + cot³θ/1 + cot²θ
Solution:
Formulae Used:
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
- 1/secθ = cosθ
- 1/cosecθ = sinθ
- a² + b² + 2ab = (a + b)²
- sin²θ + cos²θ = 1
- 2sinθcosθ = sin2θ
EXPLANATION.
⇒ sin2θ = k.
To find :
⇒ tan³θ/1 + tan²θ + cot³θ/1 + cot²θ.
As we know that,
Formula of :
⇒ 1 + tan²θ = sec²θ.
⇒ 1 + cot²θ = cosec²θ.
Put the value in the equation, we get.
⇒ tan³θ/sec²θ + cot³θ/cosec²θ.
As we know that,
Formula of :
⇒ tanθ = sinθ/cosθ.
⇒ cotθ = cosθ/sinθ.
⇒ secθ = 1/cosθ.
⇒ cosecθ = 1/sinθ.
⇒ sin³θ/cos³θ x cos²θ + cos³θ/sin³θ x sin²θ.
⇒ sin³θ/cosθ + cos³θ/sinθ.
Taking L.C.M in equation, we get.
⇒ sin⁴θ + cos⁴θ/sinθ cosθ.
We can write,
⇒ sin⁴θ + cos⁴θ = [sin²θ + cos²θ]² - 2sin²θ cos²θ.
⇒ sin⁴θ + cos⁴θ = 1 - 2sin²θ cos²θ.
Put the value in the equation, we get.
⇒ 1 - 2sin²θ cos²θ/sinθ cosθ.
⇒ 1/sinθ cosθ - 2sin²θ cos²θ/sinθ cosθ.
⇒ 1/sinθ cosθ - 2sinθ cosθ.
Put the value of sin2θ = k in equation, we get.
⇒ 1/sinθ cosθ - k.
we can multiply and divide 1/sinθ cosθ x 2, we get.
⇒ 2/2sinθ cosθ - k.
⇒ 2/k - k.
⇒ 2 - k²/k.
⇒ tan³θ/1 + tan²θ + cot³θ/1 + cot²θ = 2 - k²/k.
MORE INFORMATION.
Properties of inverse trigonometric functions.
(1) = sin⁻¹x = cosec⁻¹(1/x). = cosec⁻¹(x) = sin⁻¹(1/x).
(2) = cos⁻¹(x) = sec⁻¹(1/x). = sec⁻¹(x) = cos⁻¹(1/x).
(3) = tan⁻¹(x) = cot⁻¹(1/x). = cot⁻¹(x) = tan⁻¹(1/x).