Question

Verify the identity cot[ theta - pi/2 ] = -tan theta

Answer 1

take left hand side expression,
change it into sin and cos

Answer 2

use below :
cot = cos/sin

Answer 3

\[\cot \left[ \theta - \frac{ \pi }{ 2 } \right] = -\tan \theta\]

Where do I get cos and sin from o.o

Answer 4

^^ second reply

Answer 5

Left hand side :
\(\large \cot \left[ \theta - \frac{ \pi }{ 2 } \right]\)

\(\large \frac{\cos \left[ \theta - \frac{ \pi }{ 2 } \right]}{\sin \left[ \theta - \frac{ \pi }{ 2 } \right]}\)

Answer 6

So whenever I want to convert cot to cos and sin I just add that to the numerator and denominator?

Answer 7

by definition,
\(\large \cot x = \frac{\cos x}{\sin x}\)

Answer 8

and yes, where ever u see "cot", u can replace it wid "cos/sin"

Answer 9

next use below identities :-
cos(-x) = cos(x)
sin(-x) = -sin(x)
cos(pi/2-x) = sin(x)
sin(pi/2-x) = cos(x)

Answer 10

Left hand side :
\(\large \cot \left[ \theta - \frac{ \pi }{ 2 } \right]\)

\(\large \frac{\cos \left[ \theta - \frac{ \pi }{ 2 } \right]}{\sin \left[ \theta - \frac{ \pi }{ 2 } \right]} \)

\(\large \frac{\cos \left[ -(\frac{ \pi }{ 2 } -\theta) \right]}{\sin \left[- (\frac{ \pi }{ 2 }-\theta) \right]} \)

\(\large \frac{\cos \left[ \frac{ \pi }{ 2 } -\theta \right]}{-\sin \left[ (\frac{ \pi }{ 2 }-\theta) \right]} \)

Answer 11

fine so far ?

Answer 12

yesh...

Answer 13

good :) next use the identites :
cos(pi/2-x) = sin(x)
sin(pi/2-x) = cos(x)

Answer 14

Left hand side :
\(\large \cot \left[ \theta - \frac{ \pi }{ 2 } \right]\)

\(\large \frac{\cos \left[ \theta - \frac{ \pi }{ 2 } \right]}{\sin \left[ \theta - \frac{ \pi }{ 2 } \right]} \)

\(\large \frac{\cos \left[ -(\frac{ \pi }{ 2 } -\theta) \right]}{\sin \left[- (\frac{ \pi }{ 2 }-\theta) \right]} \)

\(\large \frac{\cos \left[ \frac{ \pi }{ 2 } -\theta \right]}{-\sin \left[ (\frac{ \pi }{ 2 }-\theta) \right]} \)

\(\large \frac{\sin \theta}{-\cos \theta} \)

\(\large -\tan \theta \)

QED

Answer 15

let me knw if domething doesnt make sense..

Answer 16

How come theta doesn't get canceled out