Question

show that
CSC 2x + (SC 4x + (50 8x+ (S0167 + CSC 32x= cota-
Cot 32​

Answer 1

LHS=cot(2x)+ CSC(2x) =Cos(2x)/sin(2x) +1/sin (2x) =1+ cos(2x) /sin(2x) =2cos^2(x)/2sin(x) cos(x) =cos(x)/sin(x) =Cot(x) proved.

Answer 2

QUESTION:

Show that Cosec 2x + Cosec 4x + Cosec 8x + Cosec 16 x + Cosec 32x = Cot x - Cot 32x.

GIVEN:

Cosec 2x + Cosec 4x + Cosec 8x + Cosec 16 x + Cosec 32x = Cot x - Cot 32x.

TO PROVE:

Cosec 2x + Cosec 4x + Cosec 8x + Cosec 16 x + Cosec 32x = Cot x - Cot 32x.

PROOF:

Take Cosec x + cot x

$\sf cosec \ x = \dfrac{1}{sin \ x}$

$\sf cot \ x = \dfrac{cos \ x}{sin\ x}$

$\sf Cosec \ \ x + cot \ x =\dfrac{1}{sin \ x}+ \dfrac{cos \ x}{sin \ x}$

$\sf Cosec \ \ x + cot \ x =\dfrac{1 + cos \ x}{sin \ x}$

$\sf Cos \ 2x =2 cos^2\ x - 1$

$\sf cos \ x = 2cos^2 \left(\dfrac{x}{2} \right) - 1$

$\sf cos \ x + 1 = 2cos^2\dfrac{x}{2}$

$\sf Sin\ 2x = 2sin\ x \ cos\ x$

$\sf Sin\ x = 2sin\ \bigg(\dfrac{x}{2} \bigg) \ cos\ \bigg(\dfrac{x}{2} \bigg)$

$\sf Cosec \ \ x + cot \ x =\dfrac{2cos^2\dfrac{x}{2}}{2sin\ \bigg(\dfrac{x}{2} \bigg) \ cos\ \bigg(\dfrac{x}{2} \bigg)}$

$\sf Cosec \ \ x + cot \ x =\dfrac{cos\dfrac{x}{2}}{sin\ \bigg(\dfrac{x}{2} \bigg)}$

$\sf Cosec \ \ x + cot \ x =cot\ \left(\dfrac{x}{2} \right)$

$\sf Cosec \ \ x =cot\ \left(\dfrac{x}{2} \right)- cot \ x$

Take Cosec 2x + cot 2x

$\sf Cosec \ 2x + cot \ 2x =\dfrac{1 + cos \ 2x}{sin \ 2x}$

$\sf cos \ 2x + 1 = 2cos^2x$

$\sf Sin\ 2x = 2sin\ x \ cos\ x$

$\sf Cosec \ 2x + cot \ 2x =\dfrac{2cos^2x}{2sin\ x \ cos\ x}$

$\sf Cosec \ 2x + cot \ 2x =\dfrac{cos \ x}{sin\ x}$

$\sf Cosec \ 2x + cot \ 2x =cot \ x$

$\sf Cosec \ 2x =cot \ x- cot \ 2x$

Similarly proceed for the next terms

$\sf cosec\ 4x = cot\ 2x - cot\ 4x$

$\sf cosec\ 8x = cot\ 4x - cot\ 8x$

$\sf cosec\ 16x = cot\ 8x - cot\ 16x$

$\sf cosec\ 32x = cot\ 16x - cot\ 32x$

Cosec 2x + Cosec 4x + Cosec 8x + Cosec 16 x + Cosec 32x = cot x - cot 2x + cot 2x - cot 4x + cot 4x - cot 8x + cot 8x - cot 16x + cot 16x - cot 32x

Cosec 2x + Cosec 4x + Cosec 8x + Cosec 16 x + Cosec 32x = cot x - cot 32x

HENCE PROVED.