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Class 12 Inverse Trigonometric Functions · Long Answer3 Marksinverse_trigonometric_identities

Class 12 Maths · Inverse Trigonometric Functions · Long Answer #63

Prove \(\sin^{-1}(8/17)+\cos^{-1}(4/5)=\cot^{-1}(36/77)\).

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Let α=arcsin(8/17) and β=arccos(4/5); both are acute. Then cos α=15/17 and sin β=3/5.
  2. Compute sin(α+β)=(8·4+15·3)/85=77/85 and cos(α+β)=(15·4−8·3)/85=36/85.
  3. Thus cot(α+β)=36/77. Also α+β∈(0,π), and its positive cosine shows that the sum is acute, within the arccot principal range.
  4. Therefore α+β=arccot(36/77), proving the identity.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let α=arcsin(8/17) and β=arccos(4/5); both are acute. Then cos α=15/17 and sin β=3/5.
  • [1 mark] Compute sin(α+β)=(8·4+15·3)/85=77/85 and cos(α+β)=(15·4−8·3)/85=36/85.
  • [1 mark] Thus cot(α+β)=36/77. Also α+β∈(0,π), and its positive cosine shows that the sum is acute, within the arccot principal range.
  • [0.5 marks] Therefore α+β=arccot(36/77), proving the identity.

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