1. y
Consider the figure beside
r y
sin
y
O x x r
y r sin 1
x
cos
r
x r cos 2
and : x 2 y 2 r 2
r cos 2 r sin 2 r 2
r 2 cos 2 r 2 sin r 2
: r2
cos 2 sin 2 1
2. cos 2 sin 2 1
cos 2 1 sin 2 The formula
sin 2 1 cos 2
y
tan
x
r sin
tan
r cos
sin
tan
cos
3. From the formula
cos 2 sin 2 1
dividing by sin 2
cos 2 1
1
sin 2 sin 2
cot an2 1 cos ec 2
cos 2 sin 2 1
dividing by cos 2
sin 2 1
1
cos cos 2
2
1 tan2 sec2
4. Examples:
1. Show that cos x 2 sin x 2 cos x sin x 5
2 2
Solution:
cos x 2 sin x2 2 cos x sin x2 5
cos 2 x 4 sin x cos x 4 sin 2 x 4 cos 2 x 4 sin x cos x sin 2 x 5
5 cos 2 x 5 sin 2 x 5
5 cos 2 x sin 2 x 5
55
5. 2. Pr ove that : sin 4 x cos 4 x 1 2 sin 2 x cos 2 x
Solution:
sin 4 x cos 4 x 1 2 sin 2 x cos 2 x
sin 2
2
x cos x 2 sin 2 x cos 2 x 1 2 sin 2 x cos 2 x
2
1 2 sin 2 x cos 2 x 1 2 sin 2 x cos 2 x
6. Activity class
Prove the following identities
sec x. cos x
1. cot x
tan x
2. 3 cos x 4 sin x 2 4 cos x 3 sin x 2 25
3. a cos x b sin x 2 b cos x a sin x 2 a 2 b 2
4. tan x cot anx cos ecx. sec x
cos x sin x 1
5.
cos x sin x cos x sin x 1 tan 2 x
6. cot B tan B cos ecB sec B