Find the value 4/3 tan^2 30 + cos^2 60 - 3 cos^2 30 - 4 tan^2 45
Answers
EXPLANATION.
Value of the equation,
⇒ 4/3 tan²(30°) + Cos²(60°) - 3Cos²(30°) - 4tan²(45°).
As we know that,
⇒ tan(30°) = 1/√3.
⇒ cos(60°) = 1/2.
⇒ cos(30°) = √3/2.
⇒ tan(45°) = 1.
Using this formula in equation, we get.
⇒ 4/3 X (1/√3)² + (1/2)² - 3(√3/2)² - 4(1)².
⇒ 4/3 X (1/3) + 1/4 - 3(3/4) - 4.
⇒ 4/9 + 1/4 - 9/4 - 4.
Taking L.C.M in equation, we get.
⇒ 16 + 9 - 81 - 144/36.
⇒ 25 - 225/36.
⇒ -200/36.
⇒ -50/9.
MORE INFORMATION.
Some useful identities.
(1) = tan(A + B + C) = ∑ tan(A) - tan(A).tan(B).tan(C)/1 - ∑ tan(A).tan(B).
(2) = tan(∅) = Cot(∅) - 2Cot(2∅).
(3) = tan3(∅) = tan∅. tan(60° - ∅). tan(60° + ∅).
(4) = tan(A + B) - tan(A) - tan(B) = tan(A). tan(B). tan(A + B).
(5) = Sin∅. Sin(60° - ∅). Sin(60° + ∅) = 1/4 Sin3∅.
(6) = Cos∅. Co(60° - ∅). Cos(60° + ∅) = 1/4 Cos3∅.
Solution :
⇒ 4/3 tan²(30°) + Cos²(60°) - 3Cos²(30°) - 4tan²(45°)
We know that
⇒ tan(30°) = 1/√3
⇒ cos(60°) = 1/2
⇒ cos(30°) = √3/2
⇒ tan(45°) = 1
On substitute values in equation, we get
⇒ 4/3 X (1/√3)² + (1/2)² - 3(√3/2)² - 4(1)²
⇒ 4/3 X (1/3) + 1/4 - 3(3/4) - 4
⇒ 4/9 + 1/4 - 9/4 - 4
Taking L.C.M in equation, we get
⇒ 16 + 9 - 81 - 144/36
⇒ 25 - 225/36
⇒ -200/36
⇒ -50/9