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Transcript 
106. If a cos  = x and b cot  = y, show that
107. If
a2 b2

1
x2 y 2
J
A
x2 y2
x
y
x
y
2
2
cos   sin   m and sin   cos   n, prove that 2  2  m  n .
a
b
a
b
a
b
108. If sec   x 
1
1
, prove that sec  + tan  = 2x or
2x
4x
109. If sec   tan   p, prove that
[CBSE 2001]
p 2 1
 sin 
p2  1
J
A
[CBSE 2004]
110. If tan  + sin  = m and tan  – sin  = n, prove that m2  n2  4 mn . [CBSE 2000, 2002(C), 2004(C)]
111. If a cos  – b sin  = c, prove that a sin   b cos    a 2  b 2  c 2
112. If 3sin   5cos   5, prove that 5sin   3cos   3
113. If sin  + sin2 = 1, prove that cos2+ cos4 = 1
[CBSE 2001(C)]
[CBSE 2002(C)]
B
114. If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A – tan A) (sec B – tan B) . (sec C – tan C), prove
that each side is equal to ± 1.
115. If x = r sin  cos , y = r sin  sin  and z = r cos , then show that x2 + y2 + z2 = r2.
HINTS TO SELECTED QUESTIONS
5sin   3cos 
5
9. here, cot   . Consider LHS =
5sin   2 cos 
4
T
I
dividing numerator and denominator by sin , we get
5  3cot 
LHS 

5  2 cot 
5
5  3 
4  7
5 2
5  2 
4
M
A
16.
tan  
1
 2. Squaring both sides,
tan 
tan 2  

1
1
 2.tan .
4
2
tan 
tan 
tan 2  
1
 4  2  2.
tan 2 
20. Consider two right triangles ACB and PRQ.
BC
QR
We have, cos B 
, cos Q 
AB
PQ
Since, cos B  cos Q 
MATHEMATICS–X
BC QR

 k , say ...(1)
AB PQ
A
C
P
B
INTRODUCTION TO TRIGONOMETRY
R
Q
147
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