Section-I

Evaluate the general solution of the trigonometric equations given below:-

i> cot2x=cosx+sinx

ii> 1-2sinx-2cosx+cotx=0 [0

iii> tanx+tan2x+tan3x=tanxtan2xtan3x

iv> cos³xsin3x+sin³xcos3x=3/4

v>4sinxcosx=1+2cosx-2sinx [0

Pls solve my problm as quickly as possible.....

Problem (i):
cot(2x) = cos(x) + sin(x)

Solution:
Express cotangent in terms of sine and cosine:
cot(2x) = cos(2x)/sin(2x)

Substitute using double-angle formulas:
cos(2x) = cos²x - sin²x
sin(2x) = 2sinx cosx

So,
cot(2x) = (cos²x - sin²x) / (2sinx cosx)

Given equation:
(cos²x - sin²x) / (2sinx cosx) = cosx + sinx

Multiply both sides by 2sinx cosx:
cos²x - sin²x = 2sinx cosx (cosx + sinx)

Expand:
cos²x - sin²x = 2sinx cosx cosx + 2sinx cosx sinx
cos²x - sin²x = 2sinx cos²x + 2sin²x cosx

Rearrange:
cos²x - 2sinx cos²x = 2sin²x cosx + sin²x
cos²x (1 - 2sinx) = sin²x (1 + 2cosx)

Divide by cos²x:
1 - 2sinx = (sin²x/cos²x) (1 + 2cosx)
1 - 2sinx = tan²x (1 + 2cosx)

This equation is nonlinear, so solving explicitly for x requires numerical methods or graphical techniques.

Problem (ii):
1 - 2sinx - 2cosx + cotx = 0

Solution:
Express cotangent in terms of sine and cosine:
cotx = cosx/sinx

So, the equation becomes:
1 - 2sinx - 2cosx + (cosx/sinx) = 0

Multiply by sinx to clear the fraction:
sinx - 2sin²x - 2sinx cosx + cosx = 0

Factorize:
sinx(1 - 2sinx - 2cosx) + cosx = 0

This equation is also nonlinear, and solving explicitly for x requires either algebraic manipulation or numerical approaches.

Problem (iii):
tanx + tan2x + tan3x = tanx tan2x tan3x

Solution:
This is a well-known identity in trigonometry:
tan(A) + tan(B) + tan(C) = tan(A) tan(B) tan(C)
when A + B + C = π

Comparing, we set:
x + 2x + 3x = π
6x = π
x = π/6 + kπ, k ∈ ℤ

Thus, the general solution is:
x = π/6 + kπ, k ∈ ℤ

Problem (iv):
cos(3x) sin(3x) + sin(3x) cos(3x) = 3/4

Solution:
Recognizing that cos(A)sin(B) + sin(A)cos(B) = sin(A + B),
we rewrite:
sin(3x + 3x) = 3/4
sin(6x) = 3/4

Solve for x:
6x = arcsin(3/4) + 2kπ
x = (1/6) arcsin(3/4) + kπ/3, k ∈ ℤ

Thus, the general solution is:
x = (1/6) arcsin(3/4) + kπ/3, k ∈ ℤ

Problem (v):
4sinx cosx = 1 + 2cosx - 2sinx

Solution:
Use the identity sin(2x) = 2sinx cosx:
2sin(2x) = 1 + 2cosx - 2sinx

Rearrange:
2sin(2x) + 2sinx - 2cosx = 1
2(sin(2x) + sinx - cosx) = 1
sin(2x) + sinx - cosx = 1/2

This equation is nonlinear, and solving explicitly for x requires numerical methods or graphical analysis.

Final Answers:
(i) The equation is nonlinear and requires numerical solving.
(ii) The equation is nonlinear and requires numerical solving.
(iii) x = π/6 + kπ, k ∈ ℤ
(iv) x = (1/6) arcsin(3/4) + kπ/3, k ∈ ℤ
(v) The equation is nonlinear and requires numerical solving.

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