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

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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

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Shaktimatha Learning | Final Boss Level Series