Problems. Your approach may be di erent but may still be ...
1. Verify the Identity:
Ans: Please note that there are more than one approach to any of the identity
problems. Your approach may be different but may still be correct.
cos2 x ? sin2 x
a.
= cot x ? tan x
cos x sin x
sin2 x
cos x sin x
cos2 x
?
=
?
= cot x ? tan x = R.H.S.
Ans: L.H.S. =
cos x sin x cos x sin x
sin x cos x
b.
cos x ? sin x
= csc x ? sec x
cos x sin x
Ans: LHS =
c.
sin x
1
1
cos x
?
=
?
= csc x ? sec x = RHS.
cos x sin x cos x sin x sin x cos x
tan x + 1
= sin x + cos x
sec x
tan x
1
1
1
sin x
Ans: LHS =
+
= tan x ¡¤
+
=
¡¤ cos x + cos x =
sec x
sec x
sec x
sec x
cos x
sin x + cos x = RHS.
d.
sin x + 1
cos x
=
cos x
1 ? sin x
Ans:
(sin x + 1)(cos x) (sin x + 1)(cos x) (sin x + 1)(cos x)
(sin x + 1)(cos x)
LHS =
=
=
=
=
(cos x)(cos x)
(cos2 x)
(1 ? sin x)(1 + sin x)
1 ? sin2 x
cos x
= RHS.
1 ? sin x
e. tan x + cot x = sec x csc x
sin x
cos x
sin2 x + cos2 x
1
Ans: LHS = tan x + cot x =
+
=
=
=
cos x
sin x
cos x sin x
cos x sin x
1
1
¡¤
= sec x csc x = RHS.
cos x sin x
sin(x + h) ? sin x
(cos h) ? 1
sin h
f.
= (sin x)
+ (cos x)
h
h
h
Ans: LHS =
sin(x + h) ? sin x sin x cos h + cos x sin h ? sin x
=
=
h
h
(sin x cos h ? sin x) + cos x cos h sin x(cos h ? 1) + cos x sin h
=
=
h
h
sin x(cos h ? 1) cos x sin h
cos h ? 1
sin h
+
= sin x ¡¤
+ cos x ¡¤
= RHS.
h
h
h
h
g.
cos x ? cos 3x
= tan x
sin x + sin 3x
x?3x
4x
sin
?2
sin
?2 sin x+3x
cos x ? cos 3x
2
2
2 sin
Ans: LHS =
=
=
sin x + sin 3x
2 sin x+3x
cos x?3x
2 sin 4x
2
2
2 cos
? sin(2x) sin(?x)
sin(2x) sin x
sin x
=
=
= tan x = RHS.
sin(2x) cos x
sin(2x) cos x cos x
h.
?2x
2
?2x
2
=
sin x
= (csc x)(1 + cos x)
1 ? cos x
sin x
(sin x)(1 + cos x)
sin x(1 + cos x) sin x(1 + cos x)
=
=
=
=
1 ? cos x (1 ? cos x)(1 + cos x)
1 ? cos2 x
sin2 x
1 + cos x
1
= (1 + cos x) ¡¤
= (1 + cos x) csc x = RHS.
sin x
sin x
Ans: LHS =
i.
sin 2x
= 1 ? cos 2x
cot x
sin 2x 2 sin x cos x
1
=
= 2 sin x cos x ¡¤
cot x
cot x
cot x
sin x
= 2 sin x cos x tan x = 2 sin x cos x ¡¤
= 2 sin2 x = sin2 x + sin2 x =
cos x
2
2
2
(1 ? cos x) + sin x = 1 ? (cos x ? sin2 x) = 1 ? cos 2x = RHS.
Ans: LHS =
j.
sin(x ? y) tan x ? tan y
=
sin(x + y) tan x + tan y
sin x cos y ? cos x sin y
(sin x cos y ? cos x sin y)/(cos x cos y)
=
=
sin x cos y + cos x sin y
(sin x cos y + cos x sin y)/(cos x cos y)
sin x cos y?cos x sin y
sin x cos y
cos x sin y
sin y
sin x
tan x ? tan y
cos x cos y
cos x cos y ? cos x cos y
cos x ? cos y
= sin x cos y cos x sin y = sin x sin y =
= RHS.
sin x cos y+cos x sin y
tan
x
+
tan
y
+
+
cos x cos y
cos x cos y
cos x cos y
cos x
cos y
Ans: LHS =
k.
2 ? sec2 x
= cos 2x
sec2 x
2
sec2 x
2 ? sec2 x
=
?
= 2 cos2 x ? 1 = cos 2x = RHS.
Ans: LHS =
2
2
2
sec x
sec x sec x
l.
sin x + sin 5x
= tan 3x
cos x + cos 5x
2 sin x+5x
sin x + sin 5x
2 cos
Ans: LHS =
=
cos x + cos 5x
2 cos x+5x
cos
2
sin 3x cos(?2x)
sin 3x
=
= tan 3x = RHS.
cos 3x cos(?2x) cos 3x
m.
6x
?4x
2 cos
2
6x
?4x
2 cos
2
sin
=
cos
x?y
2
x?y
2
cos
=?
sin
x?y
2
x?y
2
x?y
= ? cot
2
cos x ? cos y
x+y
x?y
= ? tan
tan
cos x + cos y
2
2
x?y
x+y
?2 sin x+y
sin
?
sin
cos x ? cos y
2
2
2 sin
Ans: LHS =
=
=
x+y
x?y
x+y
cos x + cos y
2 cos 2 cos 2
cos 2 cos
x+y
x?y
sin 2
sin 2
x?y
x+y
?
tan
= RHS.
x+y ¡¤
x?y = ? tan
2
2
cos 2
cos 2
x?y
2
x?y
2
sec(x) sec(y)
1 ? tan(x) tan(y)
1
1
=
=
Ans: LHS = sec(x + y) =
cos(x + y) cos x cos y ? sin x sin y
o. sec(x + y) =
1
cos x cos y
cos x cos y?sin x sin y
cos x cos y
=
1
1
1
cos x ¡¤ cos y
sin x sin y
? cos
x ¡¤ cos y
=
sec x sec y
= RHS
1 ? tan x tan y
sin(x + y)
= tan x + tan y
cos x cos y
sin(x + y) sin x cos y + cos x sin y
sin x cos y
cos x sin y
Ans: LHS =
=
=
+
=
cos x cos y
cos x cos y
cos x cos y cos x cos y
sin x sin y
+
= tan x + tan y = RHS
cos x cos y
p.
=
x?y
sin x + sin y
= ? cot
cos x ? cos y
2
2 sin x+y
sin x + sin y
2 cos
Ans: LHS =
=
cos x ? cos y
?2 sin x+y
sin
2
RHS.
n.
x?5x
2
x?5x
2
=
=
cos(x ? y)
= tan x + cot y
cos x sin y
cos(x ? y) cos x cos y + sin x sin y
cos x cos y sin x sin y
Ans: LHS =
=
=
+
=
cos x sin y
cos x sin y
cos x sin y cos x sin y
cos y sin x
+
= cot y + tan x = RHS.
sin y cos x
q.
r. sin(x + y) sin(x ? y) = cos2 y ? cos2 x
1
Ans: LHS = sin(x+y) sin(x?y) = (cos [(x + y) ? (x ? y)] ? cos [(x + y) + (x ? y)]) =
2
1
1
(cos [x + y ? x + y] ? cos [x + y + x ? y]) = (cos(2y) ? cos(2x)) =
2
2
1
1
2 cos2 (y) ? 1 ? 2 cos2 (x) ? 1 =
2 cos2 (y) ? 1 ? 2 cos2 (x) + 1 =
2
2
1 2
1
2 cos2 (y) ? 2 cos2 (x) =
2 cos (y) ? cos2 (x) = cos2 (y) ? cos2 (x) = RHS
2
2
2
1 ? tan x
s. cos(2x) =
1 + tan2 x
1 ? tan2 x 1 ? tan2 x
1
2
Ans: RHS =
=
=
(1?tan
x)¡¤
= (1?tan2 x) cos2 x =
2
2
2
1 + tan x
sec x
sec x
sin2 x
2
2
2
2
¡¤ cos2 x = cos2 x ? sin2 x = cos(2x) = LHS
cos x ? tan x cos x = cos x ?
2
cos x
t. 1 + cos(2x) = cot x sin(2x)
cos x
Ans: RHS = cot x sin(2x) =
¡¤ 2 sin x cos x = 2 cos2 x = 2 cos2 x ? 1 + 1 =
sin x
2
(2 cos x ? 1) + 1 = cos(2x) + 1 = LHS
u. sin(3x) = 3 sin x ? 4 sin3 x
Ans: LHS = sin(3x) = sin(2x + x) = sin(x) cos(2x) + cos(x) sin(2x) =
sin(x)(1?2 sin2 (x))+cos(x)¡¤2 sin(x) cos(x) = sin(x)?2 sin3 (x)+2 sin(x) cos2 (x) =
sin(x)?2 sin3 (x)+2 sin(x)(1?sin2 (x)) = sin(x)?2 sin3 (x)+2 sin(x)?2 sin3 (x) =
3 sin(x) ? 4 sin3 (x) = RHS
v. sin(2x) =
2 tan x
1 + tan2 x
1
sin x
2 tan x
2
=
2
tan
x
¡¤
=
2
tan
x
cos
x
=
2
¡¤
¡¤ cos2 x =
2
2
1 + tan x
sec x
cos x
2 sin x cos x = sin(2x) = LHS
Ans: RHS =
w. 1 + tan x tan(2x) = tan(2x) cot(x) ? 1
1
2 tan x
1
?1=
¡¤
?1=
2
tan x
1 ? tan x tan x
2
2
1 ? tan2 x 2 ? (1 ? tan2 x) 2 ? 1 + tan2 x
?1=
?
=
=
=
1 ? tan2 x
1 ? tan2 x 1 ? tan2 x
1 ? tan2 x
1 ? tan2 x
1 + tan2 x 1 + tan2 x + (tan2 x ? tan2 x) 1 + 2 tan2 x ? tan2 x 1 ? tan2 x + 2 tan2 x
=
=
=
1 ? tan2 x
1 ? tan2 x
1 ? tan2 x
1 ? tan2 x
2 tan2 x
2 tan x
1 ? tan2 x
+
=
1
+
tan
x
¡¤
= 1 + tan x tan(2x) = LHS
=
1 ? tan2 x 1 ? tan2 x
1 ? tan2 x
(cot x) ? 1 1 ? sin(2x)
x.
=
(cot x) + 1
cos(2x)
Ans: RHS = tan(2x) cot(x) ? 1 = tan(2x) ¡¤
(cot x) ? 1
Ans: LHS =
=
(cot x) + 1
cos x
sin x
cos x
sin x
x
? sin
sin x
=
x
+ sin
sin x
cos x?sin x
sin x
cos x+sin x
sin x
=
cos x ? sin x
=
cos x + sin x
(cos x ? sin x)(cos x ? sin x) cos2 x ? 2 cos x sin x + sin2 x
=
=
(cos x + sin x)(cos x ? sin x)
cos2 x ? sin2 x
cos2 x + sin2 x ? 2 cos x sin x 1 ? 2 cos x sin x 1 ? 2 sin(2x)
=
=
= RHS
cos(2x)
cos2 x ? sin2 x
cos2 x ? sin2 x
y. sin(4x) = 4 sin x cos x ? 8 sin3 x cos x
Ans: LHS = sin(4x) = sin(2(2x)) = 2 sin(2x) cos(2x) = 2(2 sin x cos x)(1 ?
2 sin2 x) = 4 sin x cos x(1 ? 2 sin2 x) = 4 sin x cos x ? 8 sin3 x cos x = RHS
z. cos(4x) = 8 cos4 x ? 8(cos2 x) + 1
2
2
2
2
Ans: LHS = cos(4x) = cos(2(2x))
=
2
cos
(2x)?1
=
2
[cos(2x)]
?1
=
2
2
cos
x
?
1
?
4
2
4
2
4
2
1 = 2 4 cos x ? 4 cos x + 1 ?1 = 8 cos x?8 cos x+2?1 = 8 cos x?8 cos x+
1 = RHS
2
cot x + tan x
2
Ans: RHS =
=
cot x + tan x
aa. sin(2x) =
cos x
sin x
2
+
sin x
cos x
=
2
cos2 x
sin x cos x
+
2
sin x
sin x cos x
=
2
cos2 x+sin2 x
sin x cos x
=
................
................
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