ట్రిగోనమెట్రీ – Step-wise Solved Problems (161–180) | Final Boss Level

161. sinθ + cosθ = a అయితే sin²θcos²θ = ?
Step: sinθcosθ = (a²−1)/2
Square → (a²−1)² / 4
Final Answer = (a²−1)² / 4

162. sinθcosθ = a అయితే sin⁸θ + cos⁸θ = ?
Step: use identity expansion
= 1 − 4a² + 2a⁴
Final Answer = 1 − 4a² + 2a⁴

163. sinθ + cosθ = √2 అయితే sin⁸θ + cos⁸θ = ?
Step: sinθcosθ = 1/2
= 1 − 4(1/4) + 2(1/16)
= 1 − 1 + 1/8
Final Answer = 1/8

164. sinθcosθ = 1/4 అయితే sin⁸θ + cos⁸θ = ?
= 1 − 4(1/16) + 2(1/256)
= 1 − 1/4 + 1/128
Final Answer = 97/128

165. sinθ + cosθ = a అయితే sin⁸θ + cos⁸θ = ?
Step: substitute sinθcosθ = (a²−1)/2
Final Answer = 1 − 4[(a²−1)/2]² + 2[(a²−1)/2]^4

166. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Final Answer = 1 − 2a²

167. sinθ + cosθ = a అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − 2[(a²−1)/2]²
Final Answer = 1 − (a²−1)²/2

168. tanθ + cotθ = a అయితే sinθcosθ = ?
Step: tanθ + cotθ = 1/(sinθcosθ)
Final Answer = 1/a

169. tanθ + cotθ = 3 అయితే sinθcosθ = ?
Step: = 1/3
Final Answer = 1/3

170. tanθ + cotθ = a అయితే tanθcotθ = ?
Step: tanθcotθ = 1
Final Answer = 1

171. sinθ + cosθ = a అయితే tanθ = ?
Step: tanθ = (sinθ)/(cosθ)
Use ratio method
Final Answer = √[(1−cos²θ)/cos²θ]

172. sinθcosθ = a అయితే tanθ + cotθ = ?
Step: = 1/a
Final Answer = 1/a

173. sinθ + cosθ = 1 అయితే sin⁸θ + cos⁸θ = ?
Step: sinθcosθ = 0
Final Answer = 1

174. sinθcosθ = 0 అయితే sin⁸θ + cos⁸θ = ?
Final Answer = 1

175. tanθ + cotθ ≥ ?
Final Answer = 2

176. sinθ + cosθ ≥ ?
Final Answer = −√2

177. sinθcosθ ≤ ?
Final Answer = 1/2

178. sin²θcos²θ ≤ ?
Final Answer = 1/4

179. sinθ + cosθ = a అయితే maximum a?
Final Answer = √2

180. sinθcosθ = a అయితే maximum a?
Final Answer = 1/2

Shaktimatha Learning | Final Boss Level Series

ట్రిగోనమెట్రీ – Step-wise Solved Problems (141–160) | Extreme Level

141. sinθ + cosθ = a అయితే sinθcosθ = ?
Step: a² = 1 + 2sinθcosθ
sinθcosθ = (a² − 1)/2
Final Answer = (a² − 1)/2

142. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Step: = (sin²θ + cos²θ)³ − 3sin²θcos²θ(sin²θ + cos²θ)
= 1 − 3a²
Final Answer = 1 − 3a²

143. sinθ + cosθ = √2 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 1/2
= 1 − 2(1/4)
Final Answer = 1/2

144. sinθ + cosθ = 1 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 0
= 1
Final Answer = 1

145. sinθcosθ = 1/4 అయితే sin⁶θ + cos⁶θ = ?
Step: = 1 − 3(1/16)
= 13/16
Final Answer = 13/16

146. sinθ + cosθ = a అయితే sin⁶θ + cos⁶θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − 3[(a²−1)/2]²
Final Answer = 1 − 3(a²−1)²/4

147. sin²θ + cos²θ + sinθcosθ = ?
Step: = 1 + sinθcosθ
Final Answer = 1 + sinθcosθ

148. sinθ + cosθ = √3 అయితే possible?
Step: max √2 < √3
Final Answer = Not possible

149. sinθcosθ = 3/4 అయితే possible?
Step: max value 1/2
Final Answer = Not possible

150. tanθ + cotθ ≥ ?
Step: ≥ 2
Final Answer = 2

151. tanθ + cotθ = 2 అయితే θ = ?
Step: tanθ = 1
Final Answer = 45°

152. tanθ + cotθ = 4 అయితే tanθ = ?
Let x = tanθ → x + 1/x = 4
Solve → x = 2 ± √3
Final Answer = 2 ± √3

153. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
= a³ − 3a[(a²−1)/2]
Final Answer = a³ − (3a(a²−1))/2

154. sinθcosθ = a అయితే sin³θcos³θ = ?
Step: = a³
Final Answer = a³

155. sinθ + cosθ = √2 అయితే sin³θcos³θ = ?
Step: sinθcosθ = 1/2
= (1/2)³
Final Answer = 1/8

156. sin²θcos²θ = 1/16 అయితే sinθcosθ = ?
Final Answer = 1/4

157. sinθ + cosθ = 0 అయితే sin³θ + cos³θ = ?
= 0 − 3×(−1/2)×0 = 0
Final Answer = 0

158. sinθ − cosθ = 1 అయితే sinθcosθ = ?
Step: square → 1 = 1 − 2sinθcosθ
sinθcosθ = 0
Final Answer = 0

159. sinθ + cosθ = 1 అయితే θ values?
Final Answer = 0° or 90°

160. sinθcosθ = 0 అయితే θ values?
Final Answer = 0° or 90°

Shaktimatha Learning | Extreme Level Practice

ట్రిగోనమెట్రీ – Step-wise Solved Problems (121–140) | Advanced Level

121. sinθ + cosθ = a అయితే sinθcosθ = ?
Step: (sinθ + cosθ)² = 1 + 2sinθcosθ
a² = 1 + 2sinθcosθ
sinθcosθ = (a² − 1)/2
Final Answer = (a² − 1)/2

122. sinθ − cosθ = a అయితే sinθcosθ = ?
Step: a² = 1 − 2sinθcosθ
sinθcosθ = (1 − a²)/2
Final Answer = (1 − a²)/2

123. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Step: = 1 − 2a²
Final Answer = 1 − 2a²

124. sinθ + cosθ = a అయితే sin²θ + cos²θ = ?
Final Answer = 1

125. sinθcosθ = a అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2a
Final Answer = 1 + 2a

126. tanθ = a అయితే sinθcosθ = ?
Step: sinθcosθ = tanθ / (1 + tan²θ)
Final Answer = a / (1 + a²)

127. cotθ = a అయితే sinθcosθ = ?
Step: = cotθ / (1 + cot²θ)
Final Answer = a / (1 + a²)

128. sinθ + cosθ = 1 అయితే sin³θ + cos³θ = ?
Step: = (sinθ + cosθ)³ − 3sinθcosθ(sinθ + cosθ)
= 1 − 3×0×1
Final Answer = 1

129. sinθ + cosθ = √2 అయితే sin³θ + cos³θ = ?
Step: = (√2)³ − 3×(1/2)×√2
= 2√2 − (3√2/2)
= √2/2
Final Answer = √2/2

130. sinθcosθ = 1/2 అయితే sin³θ + cos³θ = ?
Step: = (sinθ + cosθ)(1 − sinθcosθ)
= √2 × (1/2)
Final Answer = √2/2

131. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
Step: = a³ − 3a[(a²−1)/2]
Final Answer = a³ − (3a(a²−1))/2

132. sinθcosθ = a అయితే sin²θ + cos²θ = ?
Final Answer = 1

133. sinθ = cosθ అయితే sin³θ + cos³θ = ?
Step: θ = 45° → sin = cos = 1/√2
Answer = 2 × (1/√2)³ = √2/2
Final Answer = √2/2

134. sin²θcos²θ = a అయితే sin⁴θ + cos⁴θ = ?
Final Answer = 1 − 2a

135. sinθ + cosθ max value?
Final Answer = √2

136. sinθcosθ max value?
Final Answer = 1/2

137. sinθ + cosθ = 2 అయితే possible?
Final Answer = Not possible

138. sinθcosθ = 1 అయితే possible?
Final Answer = Not possible

139. tanθ + cotθ min value?
Step: ≥ 2
Final Answer = 2

140. tanθ + cotθ = 2 అయితే θ = ?
Step: tanθ = 1
Final Answer = 45°

Shaktimatha Learning | Advanced Problem Series

ట్రిగోనమెట్రీ – Step-wise Solved Problems (101–120) | High Level

101. sinθ + cosθ = √2 అయితే θ = ?
Step: max value occurs at 45°
Final Answer = 45°

102. sinθcosθ = 1/2 అయితే θ = ?
Step: max occurs at 45°
Final Answer = 45°

103. (sinθ + cosθ)² = 2 అయితే sinθcosθ = ?
Step: 1 + 2sinθcosθ = 2
sinθcosθ = 1/2
Final Answer = 1/2

104. sin⁴θ + cos⁴θ max value?
Step: = 1 − 2sin²θcos²θ
Max when sin²θcos²θ min (0)
Final Answer = 1

105. sin²θcos²θ max value?
Step: (sinθcosθ)² = (1/2)²
Final Answer = 1/4

106. sinθ − cosθ = 0 అయితే θ = ?
Final Answer = 45°

107. sinθ + cosθ = 0 అయితే θ = ?
Final Answer = 135°

108. tanθ = 1 అయితే sinθcosθ = ?
Step: θ = 45° → sinθcosθ = 1/2
Final Answer = 1/2

109. sinθ = 4/5 అయితే cosθ = ?
Step: √(1 − 16/25) = 3/5
Final Answer = 3/5

110. cosθ = 12/13 అయితే sinθ = ?
Step: √(1 − 144/169) = 5/13
Final Answer = 5/13

111. tanθ = 3 అయితే sinθcosθ = ?
Step: = tanθ / (1 + tan²θ)
= 3/10
Final Answer = 3/10

112. tanθ = 2 అయితే sinθcosθ = ?
Step: = 2 / (1 + 4)
Final Answer = 2/5

113. sin²θ − cos²θ = 0 అయితే θ = ?
Step: sin²θ = cos²θ
θ = 45°
Final Answer = 45°

114. sec²θ − tan²θ = ?
Final Answer = 1

115. cosec²θ − cot²θ = ?
Final Answer = 1

116. sinθcosθ = 1/4 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2(1/4) = 3/2
Final Answer = 3/2

117. sinθ + cosθ = √3 అయితే sinθcosθ = ?
Step: square → 3 = 1 + 2sinθcosθ
sinθcosθ = 1
Final Answer = 1

118. sin²θcos²θ = 1/16 అయితే sinθcosθ = ?
Final Answer = 1/4

119. sinθcosθ = 1/3 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2/3 = 5/3
Final Answer = 5/3

120. sinθ + cosθ = 2 అయితే possible?
Step: max value √2 < 2
Final Answer = Not possible

Shaktimatha Learning | High Level Practice Series

ట్రిగోనమెట్రీ – Step-wise Solved Problems (61–80) | Topper Level

61. sinθ + cosθ = √2 అయితే sinθcosθ = ?
Step 1: Square both sides
( sinθ + cosθ )² = 2
Step 2: 1 + 2sinθcosθ = 2
Step 3: sinθcosθ = 1/2
Final Answer = 1/2

62. sinθ + cosθ = 0 అయితే θ = ?
Step: sinθ = −cosθ ⇒ tanθ = −1
Final Answer = 135°

63. sin²θ + cos²θ + 2sinθcosθ = ?
Step: = (sinθ + cosθ)²
Final Answer = (sinθ + cosθ)²

64. sinθcosθ = 1/4 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2(1/4) = 1 + 1/2
Final Answer = 3/2

65. sin⁴θ + cos⁴θ value?
Step: = 1 − 2sin²θcos²θ
Final Answer = 1 − 2sin²θcos²θ

66. sinθcosθ max అయితే θ = ?
Step: max at 45°
Final Answer = 45°

67. sinθ + cosθ max అయితే θ = ?
Final Answer = 45°

68. sinθcosθ = 1/2 అయితే sinθ + cosθ = ?
Step: occurs at 45°
Final Answer = √2

69. sin²θ − cos²θ = ?
Final Answer = −cos2θ

70. tan²θ + 1 = ?
Final Answer = sec²θ

71. cot²θ + 1 = ?
Final Answer = cosec²θ

72. sec²θ − tan²θ = ?
Final Answer = 1

73. cosec²θ − cot²θ = ?
Final Answer = 1

74. sinθ = 3/5 అయితే cosθ = ?
Step: cosθ = √(1 − 9/25) = 4/5
Final Answer = 4/5

75. cosθ = 5/13 అయితే sinθ = ?
Step: sinθ = √(1 − 25/169) = 12/13
Final Answer = 12/13

76. tanθ = 4/3 అయితే sinθ = ?
Triangle method → sinθ = 4/5
Final Answer = 4/5

77. tanθ = 5 అయితే sinθcosθ = ?
Step: = tanθ / (1 + tan²θ)
= 5 / 26
Final Answer = 5/26

78. sinθcosθ = 1/2 sin2θ → prove
Identity apply
Final Answer = proved

79. sin²θcos²θ max value?
Step: (1/2)² = 1/4
Final Answer = 1/4

80. sinθ + cosθ = 1 అయితే θ = ?
Try θ = 45° → works
Final Answer = 45°

Shaktimatha Learning | Topper Level Practice

ట్రిగోనమెట్రీ – Step-wise Solved Problems (41–60)

41. (sinθ + cosθ)² + (sinθ − cosθ)² = ?
Step 1: Expand both
= (1 + 2sinθcosθ) + (1 − 2sinθcosθ)
Step 2: Add → 2
Answer = 2

42. sin⁴θ + cos⁴θ = ?
Step 1: = (sin²θ + cos²θ)² − 2sin²θcos²θ
Step 2: = 1 − 2sin²θcos²θ
Answer = 1 − 2sin²θcos²θ

43. sinθcosθ max value?
Step: occurs at θ = 45°
Answer = 1/2

44. sinθ + cosθ max value?
Answer = √2

45. sinθ + cosθ = 1 అయితే sinθcosθ = ?
Step 1: Square both sides
Step 2: 1 + 2sinθcosθ = 1
Step 3: sinθcosθ = 0
Answer = 0

46. sinθcosθ = 1/2 అయితే sinθ + cosθ = ?
Step: occurs at 45°
Answer = √2

47. sin²θ − cos²θ = ?
Answer = −cos2θ

48. 1 + tan²θ = ?
Answer = sec²θ

49. 1 + cot²θ = ?
Answer = cosec²θ

50. tanθ × cotθ = ?
Answer = 1

51. sinθ / cosθ = tanθ నిరూపించండి
Step: Definition apply
Answer = tanθ

52. cosθ / sinθ = cotθ నిరూపించండి
Answer = cotθ

53. secθ × cosθ = ?
Step: (1/cosθ) × cosθ
Answer = 1

54. cosecθ × sinθ = ?
Answer = 1

55. sinθ + cosθ min value?
Answer = −√2

56. cosθ − sinθ max value?
Answer = √2

57. tanθ period?
Answer = π

58. sinθ period?
Answer = 2π

59. cosθ period?
Answer = 2π

60. tanθ undefined value?
Answer = 90°

Shaktimatha Learning | Step-wise Practice Series

ట్రిగోనమెట్రీ – Step-wise Solved Problems (21–40)

21. sin 60° + cos 30° = ?
Step 1: sin60° = √3/2
Step 2: cos30° = √3/2
Step 3: Add → √3
Answer = √3

22. tan 60° × tan 30° = ?
Step 1: tan60° = √3
Step 2: tan30° = 1/√3
Step 3: Multiply → 1
Answer = 1

23. sin²45° + cos²45° = ?
Step 1: Identity apply
Answer = 1

24. sinθ / cosθ = ?
Step: tanθ
Answer = tanθ

25. (sinθ + cosθ)² = ?
Step 1: Formula apply
Step 2: = sin²θ + cos²θ + 2sinθcosθ
Step 3: = 1 + 2sinθcosθ
Answer = 1 + 2sinθcosθ

26. (sinθ − cosθ)² = ?
Step: = 1 − 2sinθcosθ
Answer = 1 − 2sinθcosθ

27. sinθ + cosθ = √2 అయితే θ = ?
Step: Max value occurs at 45°
Answer = 45°

28. sinθcosθ = 1/2 అయితే θ = ?
Step: Occurs at 45°
Answer = 45°

29. sin²30° + cos²60° = ?
Step 1: (1/2)² = 1/4
Step 2: (1/2)² = 1/4
Step 3: Add → 1/2
Answer = 1/2

30. tan²30° = ?
Step 1: tan30° = 1/√3
Step 2: square → 1/3
Answer = 1/3

31. sec 0° = ?
Step: 1/cos0° = 1
Answer = 1

32. cosec 90° = ?
Step: 1/sin90° = 1
Answer = 1

33. sinθ − cosθ = 0 అయితే θ = ?
Step: θ = 45°
Answer = 45°

34. tanθ = 1 అయితే θ = ?
Step: θ = 45°
Answer = 45°

35. sinθ = 1 అయితే θ = ?
Answer = 90°

36. cosθ = 0 అయితే θ = ?
Answer = 90°

37. tanθ = √3 అయితే θ = ?
Answer = 60°

38. sinθ = 1/2 అయితే θ = ?
Answer = 30°

39. cosθ = 1/2 అయితే θ = ?
Answer = 60°

40. sinθ + cosθ min value?
Answer = −√2