## anonymous one year ago Verify the identity. cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x

1. anonymous

$\cot^2x/\csc x +1 = 1 - \sin x/\sin x$

2. welshfella

try changing LHS to sine and cosines

3. anonymous

(cos^2(x)/sin^2(x))/1/sin(x) = (cos^2(x)/sin^2(x))/sin(x)

4. welshfella

just checking is the question |dw:1438357529236:dw|

5. anonymous

(cos^2(x)/sin^2(x))/1/sin(x)+sin(x)/sin(x) = (cos^2(x)/sin^2(x))/sin(x)

6. anonymous

yes

7. welshfella

ok thnx

8. welshfella

i'll write s for cos x and s for sin x |dw:1438357934242:dw| --

9. welshfella

can you continue? 2 more steps and you are home

10. anonymous

almost let me look at it

11. welshfella

(1 - s) will cancel out

12. welshfella

sorry (1 + s) will cancel out

13. anonymous

on the third step where did you get c^2/s^2

14. welshfella

c^2 / s^2 was carried on from second step

15. anonymous

so the second step was simplified to that

16. welshfella

- not very readable I know!

17. welshfella

yes

18. anonymous

ok so 1+s/s simplifies to s^2?

19. welshfella

no in the second step c^2 / s^2 is divided by (1+s) / s so we invert the (1+ s) / s to s/(1 + s) then multiply

20. anonymous

I understand now I just had to look at the equation a bit more

21. welshfella

and in 4th step we replace cos^x by 1 - sin^ x

22. welshfella

cos^2 x by 1 - sin^2 x

23. anonymous

ok

24. welshfella

this is the difference of 2 squares so simplifies to (1 - s)(1 + s)

25. anonymous

yep i understand that

26. anonymous

im ready for the next two steps

27. welshfella

so now we can cancel out (1 + s) which appears in the top and bottom of the fractions |dw:1438358863645:dw|

28. anonymous

thanks so much

29. welshfella

yw

30. anonymous

$\frac{1-\sin(x)}{\sin(x)}=\frac{1}{\sin(x)}-\frac{\sin(x)}{\sin(x)}=\csc(x)-1$ $\implies (\csc(x)-1).\frac{(\csc(x)+1)}{(\csc(x)+1)}=\frac{\csc^2(x)-1}{\csc(x)+1}=\frac{\cot^2(x)}{\csc(x)+1}$

31. anonymous

I've made use of the identities $(a-b)(a+b)=a^2-b^2$ and $1+\cot^2(x)= \csc^2(x)$