Question

cot theta - tan theta = 2 cos^2 theta-1/ sin theta cos theta

Answer 1

cot∅ - tan∅ = 2cos²∅ - 1/ sin∅ cos∅

= cos∅/sin∅ - sin∅/cos∅

= cos²∅ - sin²∅/ sin∅ cos∅

We know cos²∅ - sin²∅ = cos2∅ which is also equal to 2 cos²∅ - 1

∴ = 2cos²∅ - 1/ sin∅ cos∅

Answer 2

Answer:

We have to prove,

$cot \theta - tan \theta = \frac{2 cos^2 \theta - 1}{sin \theta. cos \theta}$,

L.H.S.

$cot \theta - tan \theta$

$=\frac{cos \theta}{sin \theta}-\frac{sin \theta}{cos \theta}$

( cot A = cos A / sin A and tan A = sin A / cos A )

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

$=\frac{cos 2\theta}{sin \theta cos \theta}$

( cos 2 A = cos² A - sin² A ),

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

( cos 2A = 2cos² A - 1 )

= R.H.S.

Hence, proved.