Question

Show that (cosec theta-cot theta) 2 =1-cos theta/ 1+cos theta

Answer 1

(cosecA-cotA)^2=1-cosA÷1+cosA
(1/sinA-cosA/sinA)^2
(1-cosA÷sinA)^2
(1-cosA)^2÷sin^2A
(1-cosA)(1-cosA)÷1-cos^2A
(1-cosA)(1-cosA)÷(1-cosA)(1+cosA)
(1-cosA)÷(1+cosA) proved

Answer 2

Answer:

We have to prove,

$(cosec \theta - cot \theta)^2 = \frac{1-cos \theta}{1+cos \theta}$

L.H.S.

$(cosec \theta - cot \theta)^2$

$=(\frac{1}{sin \theta}-\frac{cos \theta}{sin\theta})^2$

( Because, cosec A = 1/sin A and cot A = cos A/sin A )

$=(\frac{1-cos \theta}{sin \theta})^2$

$=\frac{(1-cos \theta)^2}{sin^2\theta}$

$=\frac{(1-cos \theta)^2}{1-cos^2 \theta}$

( Because, sin² A = 1 - cos² A )

$=\frac{(1-cos \theta)^2}{(1-cos \theta)(1+cos \theta)}$

$=\frac{1-cos \theta}{1+cos \theta}$

= R.H.S.

Hence, proved.