Which of the following identities, wherever defined, hold(s) good? `cotalpha-tanalpha=2cot2alpha` `tan(45^0+alpha)-tan(45^0-alpha)=2cos e c2alpha` `tan(45^0+alpha)+tan(45^0-alpha)=2sec2alpha` `tanalpha+cotalpha=2tan2alpha`

Options

A. `cot alpha-tan alpha=2cot 2alpha`
B. `tan(45^(@)+alpha)-tan(45^(@)-alpha)=2co sec 2alpha`
C. `tan(45^(@)+alpha)+tan(45^(@)-alpha)=2sec2alpha`
D. `tan alpha+cot alpha=2 tan 2alpha`.

Answer

A, C

Explanation

`(cos^(2)alpha-sin^(2)alpha)/(sinalpha cos alpha)=2cot 2alpha`
(2) `(1+t)/(1-t)(1-t)/(1+t)`, where
`t=tan alpha=((1+t)^(2)-(1-t)^(2))/(1-t^(2))`
`=(4t)/(1-t^(2))=(2xx2tanalpha)/(1-tan^(2)alpha)=2tan 2alpha`
Hence, (2) is incorrect.
(3) `(1+t)/(1-t)+(1-t)/(1+t)=((1+t)^(2)+(1-t)^(2))/(1-t^(2))`
`=(2(1+t^(2)))/(1-t^(2))`
`=(2)/(cos 2alpha)=2 sec 2alpha`
Hence, (3) is correct
(4) `tan alpha+cos alpha=(1)/(cos alpha sin alpha)=(2)/(sin 2alpha)=2co sec 2alpha`
Hence, (4) is incorrect.