(i) `4cos^(2)x=1`
(ii) `4sin^(2)x-3=0`
(iii) `tan^(2)x=1`

sin^(2) x+2sin x cos x-3cos^(2) x = 0 if (i) tan x = 3 (ii) tan x = -1 (iii) x = n pi+(pi)/(4) , n in I (iv) x = n pi+tan^(-1) (3), n in I

Solve : (i) cos 3theta + 8 cos^(3) theta = 0 (ii) "tan" (pi cot x) = cot (pi tan x) (iii) 4 cos^(2) x sin x-2 "sin"^(2) x = 3 sin x

Find the period of (i) | tan x | cos 2 x (ii) 2 sin^(4) x+ 3 cos ^(4) x (ii) | sin x| + | cos x|

Evaluate the following limits (i) lim_(x to (pi)/(2)) tan^(2) x [sqrt(2 sin^(2) x + 3 sin x + 4) - sqrt(sin^(2) x + 6 sin x + 2)] (ii) lim_(theta to 0) (sqrt(1 + sin 3 theta) -1)/(ln(1 + tan 2theta)) (iii) lim_(x to 0) (sqrt(1 + x) - ""^(3)sqrt(1 + x))/(x) (iv) lim_(phi to 0) (8)/(phi^(8)) (1 - cos (phi^(2))/(2) - cos (phi^(2))/(4) + cos (phi^(2))/(2). cos (phi^(2))/(4))

Evaluate the following limits (i) lim_(x to (pi)/(2)) tan^(2) x [sqrt(2 sin^(2) x + 3 sin x + 4) - sqrt(sin^(2) x + 6 sin x + 2)] (ii) lim_(theta to 0) (sqrt(1 + sin 3 theta) -1)/(ln(1 + tan 2theta)) (iii) lim_(x to 0) (sqrt(1 + x) - ""^(3)sqrt(1 + x))/(x) (iv) lim_(phi to 0) (8)/(phi^(8)) (1 - cos (phi^(2))/(2) - cos (phi^(2))/(4) + cos (phi^(2))/(2). cos (phi^(2))/(4))

Prove that: i) sin^(-1)(3x-4x^(3))=3sin^(-1)x, |x| le 1/2 ii) cos^(-1)(4x^(2)-3x)=3cos^(-1)x,1/2 le x le 1 iii) tan^(-1)""(3x-x^(3))/(1-3x^(2))=3tan^(-1)x, |x| lt 1/sqrt(3) iv) tan^(-1)x+tan^(-1)""(2x)/(1-x^(2))=tan^(-1)""(3x-x^(3))/(1-3x^(2))

Solve for x : i) cos(sin^(-1)x)=1/2 ii) tan^(-1)x=sin^(-1)1/sqrt(2) iii) sin^(-1)x-cos^(-1)x=pi/6

If tan x=(3)/(4) " and " pi lt x lt .(3pi)/(2) find the value of (i) sin.(x)/(2) , (ii) cos .(x)/(2) , (iii) tan .(x)/(2)