(3-4sin^2thita)(1-3tan^A)=(3-tan^2A)(4cos^2A-3)
Help please to solved this question?
I will assume there is a typo and angles are A
(3-4sin^2A)(1-3tan^A)=(3-tan^2A)(4cos^2A-3)
I find it that in most cases switching everything to sines and cosines works
(3 - 4sin^2 A)(1 - 3sin^2 A/cos^2 a) = (3 - sin^2 A/cos^2 A)(4cos^2 A - 3)
Now only cosines:
(3 - 4(1 - cos^2 A))(1 - 3(1 - cos^2 A)/cos^2 A) = (3 - (1-cos^2 A)/cos^2 A)(4cos^2 A - 3)
let x = cos^2 A
(3 - 4(1 - x) )(1 - 3(1-x)/x ) = (3 - (1 - x)/x )(4x -3)
(4x - 3)(4x - 3)/x = (4x - 1)/x ( 4x - 3)
multiply both sides by x/(4x-3)
4x-3 = 4x-1
no solution, but please check my algebra steps
It turns out to be an identity.
Both sides expand out to be
-4sin^2A - 9tan^2A + 12sin^2Atan^2A + 3
I became suspicious that it might be an identity when my last line was
(4x - 3)(4x - 3)/x = (4x - 1)/x ( 4x - 3)
my algebra error carried right down to that line
To solve the equation (3-4sin^2θ)(1-3tan^A)=(3-tan^2A)(4cos^2A-3), we can simplify the equation by using trigonometric identities and solve for the variable θ.
Step 1: Simplify the equation using trigonometric identities.
Start with the equation:
(3-4sin^2θ)(1-3tan^A) = (3-tan^2A)(4cos^2A-3)
First, recall the trigonometric identity:
tan^2A = sec^2A - 1
Applying the identity to the equation, we get:
(3-4sin^2θ)(1-3tan^A) = (3-(sec^2A - 1))(4cos^2A-3)
Simplifying further:
(3-4sin^2θ)(1-3tan^A) = (4cos^2A - sec^2A + 4cos^2A - 3)
Combine like terms:
(3-4sin^2θ)(1-3tan^A) = (8cos^2A - sec^2A - 3)
Note: The identity sec^2A = 1 + tan^2A can also be helpful.
Step 2: Expand and simplify the equation.
Expanding both sides of the equation:
3 - 9tan^A + 4sin^2θ - 12sin^2θtan^A = 8cos^2A - sec^2A - 3
Combine like terms:
-9tan^A + 4sin^2θ - 12sin^2θtan^A = 8cos^2A - sec^2A - 6
Step 3: Rearrange the equation to solve for θ.
Arrange all the terms involving θ on one side of the equation:
4sin^2θ - 12sin^2θtan^A = -9tan^A + 8cos^2A - sec^2A - 6
Combine like terms:
4sin^2θ - 12sin^2θtan^A = -9tan^A + 8cos^2A - sec^2A - 6
Step 4: Factor out sin^2θ from the left side:
sin^2θ(4 - 12tan^A) = -9tan^A + 8cos^2A - sec^2A - 6
Step 5: Divide both sides by (4 - 12tan^A):
sin^2θ = (-9tan^A + 8cos^2A - sec^2A - 6)/(4 - 12tan^A)
Now we have an expression for sin^2θ in terms of variables A.
To find the value of θ, you would need additional information about the variables A, tan^A, cos^2A, and sec^2A. Without this additional information, it is not possible to determine the exact value of θ.