4 csc 2x = 2 csc2 x tan x
I read that as
4csc(2x) = (2csc^2 x)(tanx)
LS = 4/cos (2x)
= 4/(2sinxcosx)
= 2/(sinxcosx)
RS = (2/sinx)((1/sinx)(sinx/cosx)
= 2/(sinxcosx)
= LS
To solve the equation 4 csc 2x = 2 csc 2x tan x, we need to use trigonometric identities and algebraic manipulations. Let's go step by step:
Step 1: Simplify the equation using trigonometric identities.
Recall the reciprocal identity for csc(x) which states that csc(x) = 1/sin(x).
Using this identity, we can rewrite the equation as:
4(1/sin(2x)) = 2(1/sin(2x)) * tan(x)
Step 2: Get rid of the denominators.
To eliminate the denominators, we can multiply both sides of the equation by sin(2x). This will yield:
4 = 2 * tan(x)
Step 3: Solve for tan(x).
Divide both sides of the equation by 2:
2 = tan(x)
So, the value of tan(x) is 2.
Remember that tan(x) represents the ratio of the opposite side to the adjacent side in a right triangle. Therefore, we can use the inverse tangent function (tan^(-1)) to find the value of x:
x = tan^(-1)(2)
Using a calculator or a table of trigonometric identities, we can find the value of x to be approximately 63.43 degrees or 1.107 radians.
Thus, the solution to the equation 4 csc 2x = 2 csc 2x tan x is x = 63.43 degrees or x = 1.107 radians, where csc(x) represents the cosecant function.