6tg2x+4tgx=5/cos2x.

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To solve this equation, we first need to rewrite it in terms of sine and cosine.

Recall that tangent is equal to sine divided by cosine:

Therefore, we can rewrite the equation as:

6(sin(x)/cos(x)) + 4(tan(x)) = 5/cos(2x)

Next, recall that cosine of double angle can be expressed in terms of cosine and sine using the double angle formula:

cos(2x) = cos^2(x) - sin^2(x)

Substitute this into the equation:

6(sin(x)/cos(x)) + 4(tan(x)) = 5/(cos^2(x) - sin^2(x))

Multiply through by cos(x) to eliminate the fractions:

6sin(x) + 4sin(x)cos(x) = 5

Factor out sin(x):

sin(x)(6 + 4cos(x)) = 5

Now we substitute back tan(x) = sin(x)/(cos(x) and simplify the equation:

sin(x)(6 + 4(sin(x)/cos(x))) = 5

sin(x)(6 + 4tan(x)) = 5

sin(x)(6 + 4tan(x)) = 5

6sin(x) + 4sin^2(x) = 5

Now we have a quadratic equation in terms of sin(x), which we can solve to find the possible values of x.