Sin²x + 6sinxcosx + 8cos²x=0 3tgx – 6ctgx + 7=0

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To solve the first equation:

Let sin(x) = a and cos(x) = b

The equation becomes:

a² + 6ab + 8b² = 0
(a + 2b)(a + 4b) = 0

This gives two possible solutions:
a + 2b = 0 -> sin(x) + 2cos(x) = 0
sin(x) = -2cos(x)
sin(x)/cos(x) = -2
tan(x) = -2

a + 4b = 0 -> sin(x) + 4cos(x) = 0
sin(x) = -4cos(x)
sin(x)/cos(x) = -4
tan(x) = -4

To solve the second equation:

3tan(x) - 6cot(x) + 7 = 0
3tan²(x) - 6 + 7tan(x) = 0
3(tan²(x) + 7tan(x) - 2) = 0

This is a quadratic equation in terms of tan(x) and can be solved using the quadratic formula:

tan(x) = (-b ± √(b² - 4ac)) / 2a

Plugging in the values:
a = 1, b = 7, c = -2

tan(x) = (-7 ± √(7² - 4(1)(-2))) / 2(1)
tan(x) = (-7 ± √(49 + 8)) / 2
tan(x) = (-7 ± √57) / 2

Therefore, the solutions for the given equations are tan(x) = -2, tan(x) = -4, and tan(x) = (-7 ± √57) / 2.