Question

If sinA + cosecA = 2, then sin⁸A + cosec⁸A =

Answer 1

Answer :

sin⁸A + cosec⁸A = 2

Solution :

• Given : sinA + cosecA = 2
• To find : sin⁸A + cosec⁸A = ?

We have ;

sinA + cosecA = 2

Now ,

Squaring both the sides , we get ;

=> (sinA + cosecA)² = 2²

=> sin²A + 2•sinA•cosecA + cosec²A = 4

=> sin²A + 2•sinA•(1/sinA) + cosec²A = 4

=> sin²A + 2 + cosec²A = 4

=> sin²A + cosec²A = 4 - 2

=> sin²A + cosec²A = 2

Again squaring both the sides , we get ;

=> (sin²A + cosec²A)² = 2²

=> (sin²A)² + 2•sin²A•cosec²A + (cosec²A)² = 4

=> sin⁴A + 2•sin²A•(1/sin²A) + cosec⁴A = 4

=> sin⁴A + 2 + cosec⁴A = 4

=> sin⁴A + cosec⁴A = 4 - 2

=> sin⁴A + cosec⁴A = 2

Again squaring both the sides , we get ;

=> (sin⁴A + cosec⁴A)² = 2²

=> (sin⁴A)² + 2•sin⁴A•cosec⁴A + (cosec⁴A)² = 4

=> sin⁸A + 2•sin⁴A•(1/sin⁴A) + cosec⁸A = 4

=> sin⁸A + 2 + cosec⁸A = 4

=> sin⁸A + cosec⁸A = 4 - 2

=> sin⁸A + cosec⁸A = 2

Hence ,

sin⁸A + cosec⁸A = 2