3. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
4. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
5. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t
6. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t ;
2
tanIf
t
2
1
2
tan
t
t
7. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t ;
2
tanIf
t
2
1
2
tan
t
t
2
1
2
sin
t
t
8. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t ;
2
tanIf
t
2
1
2
tan
t
t
2
1
2
sin
t
t
2
2
1
1
cos
t
t
9. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t ;
2
tanIf
t
2
1
2
tan
t
t
2
1
2
sin
t
t
2
2
1
1
cos
t
t
Note:
If tan ;t 2
2
tan 2
1
t
t
10. The t results
Let tan
2
t
2
2tan
2tan
1 tan
2
2
1
2
tan
t
t
t2
2
1 t
2
1 t
222 2
2 1h t t
2 2 4
4 1 2t t t
4 2
2 1t t
22
1t ;
2
tanIf
t
2
1
2
tan
t
t
2
1
2
sin
t
t
2
2
1
1
cos
t
t
Note:
If tan ;t 2
2
tan 2
1
t
t
If tan 2 ;t 2
2
tan 4
1
t
t
11.
1 cos
e.g. i Show that , where tan
sin 2
x x
t t
x
12.
1 cos
e.g. i Show that , where tan
sin 2
x x
t t
x
1 cos
sin
x
x
2
2
2
1
1
1
2
1
t
t
t
t
is double ,
2
so results can be used for sin ,cos
x
x
t x x
13.
1 cos
e.g. i Show that , where tan
sin 2
x x
t t
x
1 cos
sin
x
x
2
2
2
1
1
1
2
1
t
t
t
t
is double ,
2
so results can be used for sin ,cos
x
x
t x x
2 2
1 1
2
t t
t
14.
1 cos
e.g. i Show that , where tan
sin 2
x x
t t
x
1 cos
sin
x
x
2
2
2
1
1
1
2
1
t
t
t
t
is double ,
2
so results can be used for sin ,cos
x
x
t x x
2 2
1 1
2
t t
t
2
2
2
t
t
t
16. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
17. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
18. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
19. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
20. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
21. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
22. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
Let tan ;
2
t
1 sin cos
1 sin cos
2
2 2
2
2 2
2 1
1
1 1
2 1
1
1 1
t t
t t
t t
t t
23. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
Let tan ;
2
t
1 sin cos
1 sin cos
2
2 2
2
2 2
2 1
1
1 1
2 1
1
1 1
t t
t t
t t
t t
2 2
2 2
1 2 1
1 2 1
t t t
t t t
24. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
Let tan ;
2
t
1 sin cos
1 sin cos
2
2 2
2
2 2
2 1
1
1 1
2 1
1
1 1
t t
t t
t t
t t
2 2
2 2
1 2 1
1 2 1
t t t
t t t
2
2 2
2 2
t t
t
25. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
Let tan ;
2
t
1 sin cos
1 sin cos
2
2 2
2
2 2
2 1
1
1 1
2 1
1
1 1
t t
t t
t t
t t
2 2
2 2
1 2 1
1 2 1
t t t
t t t
2
2 2
2 2
t t
t
2 1
2 1
t t
t
26. 2
2tan75
ii Use tan to simplify
2 1 tan 75
t
Let tan75 ;t
i.e. 75
2
150
2
2tan75
1 tan 75
2
2
1
t
t
sin
sin150
1
2
1 sin cos
iii Prove tan
1 sin cos 2
Let tan ;
2
t
1 sin cos
1 sin cos
2
2 2
2
2 2
2 1
1
1 1
2 1
1
1 1
t t
t t
t t
t t
2 2
2 2
1 2 1
1 2 1
t t t
t t t
2
2 2
2 2
t t
t
2 1
2 1
t t
t
t tan
2
27. 2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,
2
show that cosec cot cot
2
t
28. 2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,
2
show that cosec cot cot
2
t
cosec cot
2 2
1 1
2 2
t t
t t
29. 2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,
2
show that cosec cot cot
2
t
cosec cot
t
t
1
2
2
2 2
1 1
2 2
t t
t t
30. 2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,
2
show that cosec cot cot
2
t
cosec cot
t
t
1
2
2
2
cot
2 2
1 1
2 2
t t
t t
31. 2005 Extension 1 HSC Q4b)
(iv) By making the substitution tan or otherwise,
2
show that cosec cot cot
2
t
cosec cot
t
t
1
2
2
2
cot
2 2
1 1
2 2
t t
t t
Exercise 2B; 1def, 3ace, 4bcf, 5aceg, 6, 8bd, 10a
