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Answer 1

1.
sin²(x) + cos²(x) = (√3 + 1)•sin(x)•cos(x)
2•sin(x)•cos(x) = sin(2•x) => sin(x)•cos(x) =  (sin(2•x))/2
sin²(x) = 1 - cos²(x)
1 - cos²(x) + cos²(x) = (√3 + 1)•(sin(2•x))/2
1 = (√3 + 1)•(sin(2•x))/2
2 = (√3 + 1)•sin(2•x)
sin(2•x) = 2/(√3 + 1)
2•x = arcsin(2/(√3 + 1)) + 2•π•n,    n ∈ Z
x = 0,5•arcsin(2/(√3 + 1)) + π•n,    n ∈ Z

2.

sin²x + √3cos²x = (√3 + 1)sinx•cosx

sin²x/cos²x + √3cos²x/cos²x = (√3 + 1)sinx•cosx/cos²x

tg²x + √3 = (√3 + 1)tgx

tg²x + √3 = √3tgx + tgx

tgx = y

y² + √3 = y√3 + y

y(y -√3) = y - √3

y = 1

y = √3

tgx = 1 =>  x = π/4 + πn,    n ∈ Z

tgx = √3 => x= π/3 + πn,    n ∈ Z