Answer:
Using the identity cot(x) = 1/tan(x), we can rewrite the given equation as:
1/tan(x - 10)° = tan (4x)°
Next, we can use the identity tan(-x) = -tan(x) to rewrite the right-hand side as:
tan(-4x)° = -tan(4x)°
Substituting this back into the equation, we get:
1/tan(x - 10)° = tan(-4x)°
Multiplying both sides by tan(x - 10)°, we get:
tan(-4x)° * tan(x - 10)° = 1
Using the identity tan(-x) = -tan(x), we can rewrite the left-hand side as:
-tan(4x)° * tan(x - 10)° = -1
Now, we can use the identity tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)*tan(b)) to rewrite the left-hand side as:
-(tan(4x)° - tan(x - 10)°)/(1 + tan(4x)° * tan(x - 10)°) = -1
Multiplying both sides by the denominator, we get:
tan(4x)° - tan(x - 10)° = 1 + tan(4x)° * tan(x - 10)°
Using the identity tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)*tan(b)), we can rewrite the left-hand side as:
tan(4x + (10 - x))°/(1 - tan(4x)° * tan(x - 10)°) = 1 + tan(4x)° * tan(x - 10)°
Simplifying the expression on the left-hand side, we get:
tan(3x + 10)°/(1 - tan(4x)° * tan(x - 10)°) = 1 + tan(4x)° * tan(x - 10)°
Multiplying both sides by the denominator, we get:
tan(3x + 10)° = (1 - tan(4x)° * tan(x - 10)°) + (tan(4x)° * tan(x - 10)°) * (1 - tan(4x)° * tan(x - 10)°)
Expanding and simplifying the right-hand side, we get:
tan(3x + 10)° = 1 - tan(4x)° * tan(x - 10)° + tan(4x)° * tan(x - 10)° - tan^2(4x)° * tan^2(x - 10)°
Simplifying further, we get:
tan(3x + 10)° = 1 - tan^2(4x)° * tan^2(x - 10)°
Using the identity tan^2(x) = sec^2(x) - 1, we can rewrite the right-hand side as:
tan(3x + 10)° = sec^2(4x)° * sec^2(x - 10)° - 1
Now, we can use the fact that sec(x) = 1/cos(x) and simplify the right-hand side further:
tan(3x + 10)° = (1/cos^2(4x)°) * (1/cos^2(x - 10)°) - 1
Multiplying both sides by cos^2(4x)° * cos^2(x - 10)°, we get:
tan(3x + 10)° * cos^2(4x)° * cos^2(x - 10)° = 1 - cos^2(4x)° * cos^2(x - 10)°
Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the right-hand side as:
tan(3x + 10)° * cos^2(4x)° * cos^2(x - 10)° = sin^2(4x)° * sin^2(x - 10)°
Now, we can use the identity sin(2x) = 2sin(x)cos(x) to rewrite the left-hand side as:
2tan(3x + 10)° * cos(4x)° * cos(x - 10)° * sin(4x)° * sin(x - 10)° = sin^2(4x)° * sin^2(x - 10)°
Dividing both sides by sin^2(4x)° * sin^2(x - 10)°, we get:
2tan(3x + 10)° * cos(4x)° * cos(x - 10)° = 1
Using the identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b), we can rewrite the left-hand side as:
2tan(3x + 10)° * (cos(4x)° * cos(x)° + sin(4x)° * sin(x)°) * (cos(x)° * cos(10)° + sin(x)° * sin(10)°) = 1
Simplifying the expression, we get:
2tan(3x + 10)° * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + cos(4x)° * sin(x)° * sin(10)° + sin(4x)° * cos(x)° * sin(10)°) = 1
Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the last two terms as:
2tan(3x + 10)° * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + sin(x + 4x)° * sin(10)°) = 1
Simplifying the expression, we get:
2tan(3x + 10)° * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + sin(5x)° * sin(10)°) = 1
Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) again, we can rewrite the last term as:
2tan(3x + 10)° * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + cos(5x - 10)° * sin(10)°) = 1
Now, we can use the fact that tan(x) = sin(x)/cos(x) to rewrite the left-hand side as:
2(sin(3x + 10)°/cos(3x + 10)°) * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + cos(5x - 10)° * sin(10)°) = 1
Multiplying both sides by cos(3x + 10)°, we get:
2sin(3x + 10)° * (cos(4x)° * cos(x)° * cos(10)° + sin(4x)° * sin(x)° * cos(10)° + cos(5x - 10)° * sin(10)°) = cos(3x + 10)°
Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) again, we can rewrite the expression in the parentheses as:
cos(10)° * (cos(4x - x)° * sin(10)° + sin(4x)° * cos(x - 10)°) + sin(5x - 10)° * sin(10)°
Simplifying further, we get:
cos(10)° * (cos(3x)° * sin(10)° + sin(3x)° * cos(10)°) + sin(5x - 10)° * sin(10)°
Now, we can use the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) one more time to rewrite the last term as:
sin(5x)° * cos(10)° * sin(10)° - cos(5x)° * sin(10)° * cos(10)°
Substituting all these expressions back into the original equation, we get:
2sin(3x + 10)° * (cos(3x)° * sin(10)° + sin(3x)° * cos(10)° + cos(5x)° * sin(10)° * cos(10)° - sin(5x)° * cos(10)° * sin(10)°) = cos(3x + 10)° / cos(10)°
2sin(3x + 10)° * (cos(3x)° * sin(10)° + sin(3x)° * cos(10)° + cos(5x)° * sin(10)° * cos(10)° - sin(5x)° * cos(10)° * sin(10)°) = cos(3x + 10)° / cos(10)°
Multiplying both sides by cos(10)°, we get:
2sin(3x + 10)° * (cos(3x)° * sin(10)° + sin(3x)° * cos(10)° + cos(5x)° * sin(10)° * cos(10)° - sin(5x)° * cos(10)° * sin(10)°) * cos(10)° = cos(3x + 10)°