i don't know how to solve this, i'm so lost.

determine the exact value of c

cot( 4c- pie/4) + tan ( 2c + pie/4)= 0

please help me!!!

THANK YIU

cot(4c-π/4) = 1/tan(4c-π/4)

= 1/[ (tan4c) - tanπ/4)/(1 + tan4c(tanπ/4))]
= (1+tan4ctanπ/4)/(tan4c - tanπ/4)
= (1 + tan4c)/(tan4c - 1) , since tan π/4 = 1

also from tan 2A = 2tanA/(1 - tan^2 A)
the above is
= [ 1 + 2tan2c/(1 - tan^2 2c) ] / [2tan2c/(1 - tan^2 2c) - 1 ] ....ARGGGG!
let tan 2c = x
then above
= ([ 1 + 2x/(1-x^2) ] / [2x/(1-x^2) - 1 ]
= (1-x^2+2x)/(2x-1+x^2)

similarly tan(2c + /4)
= (tan2c + tan π/4) / (1 - tan2ctanP/4)
= (x+1)/(1-x)

original equation becomes
(1-x^2+2x)/(2x-1+x^2) = -(x+1)/(1-x)
cross-multiply .... and simplify

2x^3 + 2x = 0
2x(x^2 + 1) = 0
x = 0 or x^2+1=0
no solution for the second part, so

x = 0
tan 2c = 0
2c = 0 , 180, 360 , 540, 720..

c = 0, 90, 180, 270 , 360 , ... (degrees)

(somehow I think there is an easier way)

To determine the exact value of c in the equation cot(4c - pi/4) + tan(2c + pi/4) = 0, we need to simplify the equation.

First, let's rewrite the tangent function:
tan(x) = sin(x) / cos(x)

Now, substitute the simplified tangent expression into the given equation:
cot(4c - pi/4) + sin(2c + pi/4) / cos(2c + pi/4) = 0

Next, simplify the cotangent function:
cot(x) = cos(x) / sin(x)

Now, substitute the simplified cotangent expression into the equation:
cos(4c - pi/4) / sin(4c - pi/4) + sin(2c + pi/4) / cos(2c + pi/4) = 0

To simplify further, we need to find a common denominator for the fractions:
(cos(4c - pi/4) × cos(2c + pi/4)) / (sin(4c - pi/4) × cos(2c + pi/4)) + (sin(2c + pi/4) × sin(4c - pi/4)) / (cos(2c + pi/4) × sin(4c - pi/4)) = 0

Combine the fractions:
[cos(4c - pi/4) × cos(2c + pi/4) + sin(2c + pi/4) × sin(4c - pi/4)] / [sin(4c - pi/4) × cos(2c + pi/4)] = 0

Apply the trigonometric identity:
cos(A - B) = cos(A) × cos(B) + sin(A) × sin(B)

The equation becomes:
cos[4c - (pi/4 - (2c + pi/4))] / [sin(4c - pi/4) × cos(2c + pi/4)] = 0

Simplify the angle in the cosine function:
cos[4c - (pi/4 - (2c + pi/4))] / [sin(4c - pi/4) × cos(2c + pi/4)] = 0

Simplify further:
cos(4c - 2c) / [sin(4c - pi/4) × cos(2c + pi/4)] = 0

cos(2c) / [sin(4c - pi/4) × cos(2c + pi/4)] = 0

Now, we have a fraction equal to zero. For a fraction to equal zero, the numerator must be equal to zero:
cos(2c) = 0

To find the values of c that satisfy this equation, we look at the unit circle. The cosine function is equal to zero at multiples of pi/2:
2c = pi/2 + k*pi, where k is an integer

Divide both sides by 2:
c = (pi/4 + k*pi/2), where k is an integer

Therefore, the exact values of c that satisfy the given equation are c = pi/4 + k*pi/2, where k is an integer.

To solve the equation cot(4c - pi/4) + tan(2c + pi/4) = 0 and find the exact value of c, you can use trigonometric identities and algebraic methods. Here's how you can approach it step by step:

Step 1: Recognize the trigonometric identities:
cot(x) = 1/tan(x)
tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))

Step 2: Rewrite the equation:
cot(4c - pi/4) + tan(2c + pi/4) = 0
1/tan(4c - pi/4) + (tan(2c) + tan(pi/4))/(1 - tan(2c)tan(pi/4)) = 0

Step 3: Simplify the equation:
1/tan(4c - pi/4) + (tan(2c) + 1)/(1 - tan(2c)) = 0
[(1 - tan(2c))/(tan(4c - pi/4)(1 - tan(2c)))] + [(tan(2c) + 1)/(1 - tan(2c))] = 0

Step 4: Find a common denominator and simplify further:
[(1 - tan(2c)) + tan(4c - pi/4)(tan(2c) + 1)] / [tan(4c - pi/4)(1 - tan(2c))] = 0

Step 5: Combine terms and simplify:
(1 - tan(2c) + tan(4c - pi/4)tan(2c) + tan(4c - pi/4)) / [tan(4c - pi/4)(1 - tan(2c))] = 0

Step 6: Apply difference of squares identity and simplify:
[(1 - tan(2c))(1 + tan(4c - pi/4))] / [tan(4c - pi/4)(1 - tan(2c))] = 0

Step 7: Set each factor equal to zero:
1 - tan(2c) = 0 or 1 + tan(4c - pi/4) = 0

Step 8: Solve each equation separately:
For 1 - tan(2c) = 0:
tan(2c) = 1
2c = pi/4 + n*pi (where n is an integer)
c = (pi/4 + n*pi)/2 (where n is an integer)

For 1 + tan(4c - pi/4) = 0:
tan(4c - pi/4) = -1
4c - pi/4 = 3pi/4 + n*pi (where n is an integer)
4c = pi + n*pi (where n is an integer)
c = (pi + n*pi)/4 (where n is an integer)

So the exact values for c are c = (pi/4 + n*pi)/2 and c = (pi + n*pi)/4, where n is an integer.

Please note that these are general solutions, and you might need to further verify and restrict the domain depending on the given conditions or context of the problem.