5(tan^2x-cos^2x)=2cos2x+9 find cos4x

Question

5(tan^2x-cos^2x)=2cos2x+9  find cos4x

Answer

5(sec^2 x – 1 – cos^2 x) = 2(2cos^2 x – 1) + 9Let cos^2 x = t5(1/t - 1 - t) = 2(2t - 1) + 95(1 – t – t^2) = 4t^2 + 7t9t^2 + 12t – 5 = 0t = 1/3, -5/3cos^2 x = 1/3cos 2x = 2/3 - 1 = -1/3cos 4x = -7/9

Answer

5 (tan^2x - cos^2x) = 2cos2x +95 ( (sec^2x - 1) - cos^2x) = 2(2cos^2x - 1) +95 (1/cos^2x - 1- cos^2x) = 4cos^2x - 2 + 95 (1 - cos^2x - cos^4x ) = 4cos^4x + 7cos^2xYou solve this we get = 9cos^4x+12cos^2x-5Therefore let cos^4x =y^2 and cos^2x = y.0=9y^2 + 12y - 5You solve this we get answer.

Answer

Given Equation:
5(tan^2x-cos^2x)=2cos2x+9

Find cos4x:
To find cos4x, we need to simplify the given equation and express it in terms of cos4x.

Step 1: Simplify the Equation
Start by expanding the left side of the equation using trigonometric identities:
5(tan^2x - cos^2x) = 5((sin^2x/cos^2x) - cos^2x) = 5(sin^2x - cos^4x)

Step 2: Use Double Angle Identity
Now, utilize the double angle identity for cosine:
cos2x = 2cos^2x - 1
cos^2x = (1 + cos2x)/2
Substitute cos^2x back into the equation:
5(sin^2x - (1 + cos2x)/2) = 5sin^2x - 5/2 - 5cos2x

Step 3: Combine Terms
Combine like terms on the right side of the equation:
5sin^2x - 5/2 - 5cos2x = 2cos2x + 9

Step 4: Solve for cos4x
Now, we have the equation in terms of sin^2x and cos2x. To find cos4x, we need to express sin^2x in terms of cos2x using the Pythagorean Identity: sin^2x = 1 - cos^2x
Substitute sin^2x = 1 - cos^2x back into the equation and simplify:
5(1 - cos^2x) - 5/2 - 5cos2x = 2cos2x + 9
5 - 5cos^2x - 5/2 - 5cos2x = 2cos2x + 9
-5cos^2x - 5cos2x - 5/2 = 2cos2x + 4
Finally, we have the expression for cos4x in terms of cos2x:
cos4x = -5cos^2x - 5cos2x - 5/2 - 2cos2x - 4

Answer

-7/9

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