Question

prove that cos\1-tan +sin\1-cot= sin+cos

Answer 1

the answer is given in the picture.

Answer 2

To Prove:

cos\1-tan + sin\1-cot= sin+cos

Solution :

L.H.S.=

$\frac{cosA}{1-tanA}+ \frac{sinA}{1- cotA} (tanA\frac{sinA}{cosA})\ (cot= \frac{cosA}{sinA)\\}$

$\frac{cosA}{1-sinA/cosA} + \frac{sinA}{1-cosA/sinA}$

$\frac{cos^{2}A }{cosA-sinA} + \frac{sin^{2}A }{sinA-cosA}\\\frac{cos^{2}A }{cosA-sinA} - \frac{sin^{2} A}{cosA-sinA} \\\frac{cos^{2}A-sin^{2}A }{cosA-sinA}$(a²-b²= (a+b) (a-b)

$\frac{(cosA+sinA)(cosA-sinA)}{(cosA-sinA)}$

sinA + cosA = R.H.S

Hence, proved.