**Find the principal and general solutions of the equation tanx = √3**

Solution:

tanx = √3

We know that,

tan ^{π}/_{3} = √3 and tan^{4}^{π}/_{3} = tan(π + ^{π}/_{3}) = tan^{ π}/_{3 }= √3

therefore, the principal solutions are x = ^{π}/_{3} and ^{4}^{π}/_{3}

Now, tanx = tan ^{π}/_{3}

x =n π+ ^{π}/_{3}, where n € Z

Therefore, the general solution is x = n π +^{ π}/_{3}, where n € Z

**Find the principal and general solutions of the equation secx = 2**

Solution:

secx = 2

We know that sec^{π}/_{3} = 2 and sec ^{5}^{π}/_{3} = sec(2π – ^{π}/_{3}) =sec^{π}/_{3 }= 2

therefore, the principal solutions are x = ^{π}/_{3} and ^{5}^{π}/_{3}

Now, sec = sec^{π}/_{3}

cosx = cos^{π}/_{3}

x = 2nπ± ^{π}/_{3}, where n € Z

Thus, the general so;lution is x = 2nπ± ^{π}/_{3}, where n € Z

**Find the principal and general solutions of the equation cotx = -√3**

Solution:

cot x = -√3

We know that cot ^{π}/_{6} =√3

cot(π – ^{π}/_{6}) = -cot^{π}/_{6} = -√3 and cot(2π – ^{π}/_{6}) = -cot^{π}/_{6} = -√3

i.e., cot ^{5}^{π}/_{6} = -√3

and cot ^{11}^{π}/_{6} = -√3

Thus the principal solutions are x = ^{5}^{π}/_{6} and ^{11}^{π}/_{6}

Now, cotx = cot ^{5}^{π}/_{6}

tanx = tan ^{5}^{π}/_{6} [cot x = ^{1}/_{tanx}}

x = nπ +^{5π}/_{6} , where n €Z

thus, the general solution is x = nπ + ^{5π}/_{6}, where n €Z

**find the general solution of cosecx = -2**

solution:

We know that cosec ^{π}/_{6} = 2

cosec(π + ^{π}/_{6}) = -cosec^{π}/_{6} = -2

and cosec(2π-^{π}/_{6}) = -cosec^{π}/_{6} = -2

i.e., cosec^{7}^{π}/_{6} = -2 and cosec^{11}^{π}/_{6} = -2

Therefore the principal solutions are x = ^{7}^{π}/_{6} and ^{11}^{π}/_{6}

Now, cosecx = cosec^{7}^{π}/_{6}

sinx = sin^{7}^{π}/_{6} [cosecx = ^{1}/_{sinx}]

x = nπ + (-1)^{n }^{7}^{π}/_{6}, where n €Z

Therefore, the general solution is x = nπ + (-1)^{n }^{7}^{π}/_{6}, where n €Z

**Find the general solution of the equation cos4x = cos 2x**

Solution:

cos4x = cos2x

cos4x – cos2x = 0

-2sin(^{4x+2x}/_{2})sin(^{4x-2x}/_{2})=0

[cosA-cosB = -2sin(^{A+B}/_{2})sin(^{A-B}/_{2})]

sin3x.sinx = 0

sin3x = 0 or sinx =0

3x = nπ or x = nπ, where n €Z

x = ^{nπ}/_{3} or x = nπ, where n €Z

**Find the general solution of the equation cos3x+cosx-cos2x = 0**

Solution:

cos3x + cosx – cos2x = 0

2cos(^{3x+x}/_{2})cos(^{3x-x}/_{2}) – cos2x = 0

[cosA +cosB = 2cos(^{A+B}/_{2})cos(^{A-B}/_{2})]

2cos2x.cosx – cos2x =0

cos2x(2cosx – 1) = 0

cos 2x =0 or 2cosx-1 = 0

cos2x = 0 or cosx = ^{1}/_{2}

2x = (2n+1)^{ π}/_{2} or cosx = cos^{π}/_{3}, where n €Z

x = (2n+1) ^{π}/_{3 } or x = 2nπ±^{π}/_{3}, where n €Z

**Find the general solution of the equation sin2x + cosx = 0**

Solution:

sin 2x + cos x = 0

2sinxcosx + cosx = 0

cosx(2sinx + 1) = 0

cosx = 0 or 2sinx+1=0

Now, cosx = 0 ; cosx = (2n+1)^{ π}/_{2} , where n €Z

2sinx + 1 = 0

sinx = –^{1}/_{2} = -sin^{π}/_{6} = sin(π+^{π}/_{6}) = sin ^{7}^{π}/_{6}

x = nπ + (-1)^{n} ^{7π}/_{6}, where n €Z

Therefore, the general solution is (2n+1) ^{π}/_{2} or nπ+(-1)^{n} ^{7π}/_{6}, n €Z

- Find the general solution of the equation sec
^{2}2x = 1 – tan2x

Solution:

sec^{2}2x = 1 – tan2x

1+tan^{2}2x = 1 – tan2x

tan^{2}2x + tan2x = 0

tan 2x(tan2x+1) = 0

tan 2x = 0 or tan2x+1 = 0

Now, tan 2x = 0

tan 2x = tan0

2x = nπ+0, where n €Z

tan2x + 1 = 0

tan 22x = -1 = -tan^{π}/_{4} = tan(π-^{π}/_{4}) = tan ^{3}^{π}/_{4}

2x = nπ + ^{3π}/_{4}, where n €Z

x = ^{n}^{π}/_{2} + ^{3π}/_{8}, where n €Z

Therefore the general solution is ^{nπ}/_{2} or ^{n}^{π}/_{2} + ^{3π}/_{8}, where n €Z

**Find the general solution of the equation sinx + sin3x+ sin5x = 0**

Solution:

sinx + sin3x+ sin5x = 0

(sinx + sin5x) + sin3x = 0

[2sin(^{x+5x}/_{2})cos(^{x-5x}/_{2})]+ sin3x = 0

sinA+sinB = 2sin(^{A+B}/_{2})cos(^{A-B}/_{2})]

2sin3xcos(-2x)+sin3x = 0

2sin3x cos2x + sin3x = 0

sin3x(2cos2x+1)=0

sin33x = 0 or 2cos2x+1 =0

Now, sin3x = 0 ; 3x = n π , where n €Z

i.e., x = ^{n π}/_{3}, where n €Z

2cos2x + 1 = 0

cos2x = –^{1}/_{2} = -cos^{ π}/_{3} = cos(π-^{ π}/_{3})

cos2x = cos^{2}^{ π}/_{3}

2x = 2n π ±^{ 2π}/_{3}, where n €Z

x = n π ±^{ π}/_{3}, where n €Z

Therefore the general solution is nπ or n π ±^{ π}/_{3}, where n €Z