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AP EAMCET20224 Jul 2022Morning ShiftMathematicsTrigonometric Ratios & IdentitiesActual

If 1 + tan 1 ° 1 + tan 2 ° … 1 + tan 45 ° = 2 n , then n =

Options

  1. A0
  2. B32
  3. C23
  4. D2

Correct answer

C. 23

Step-by-step solution

Consider A + B = 45 ° ⇒ tan A + B = 1 ⇒ tan A + tan B = 1 - tan A tan B ⇒ 1 + tan A + tan B + tan A tan B = 2 ⇒ ( 1 + tan   A ) ( 1 + tan   B ) = 2 So, 1 + tan 1 ° 1 + tan 44 ° = 1 + tan 2 ° 1 + tan 43 ° = . . . . = 1 + tan 22 ° 1 + tan 23 ° = 2 Thus, 1 + tan 1 ° 1 + tan 2 ° . . . . . . 1 + tan 43 ° 1 + tan 44 ° = 2 22 Now, 1 + tan 45 ° = 1 + 1 = 2 So, ( 1 + tan   1 o ) 1 + tan   2 o … . 1 + tan   44 &

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