NCERT Class 12 · Ch 2 Miscellaneous Exercise · Q6
NCERT Class 12 Maths · Chapter 2 Inverse Trigonometric Functions · Miscellaneous Exercise Question 6
Prove cos⁻¹(12/13)+sin⁻¹(3/5)=sin⁻¹(56/65).
💡 Concept Hints
- Recall the principal range of the inverse function before simplifying.
- Find an angle with the required trigonometric value, then verify it belongs to the principal range.
✅ Step-by-Step Solution
Final Answer: The identity holds.
- Let α=cos⁻¹(12/13), β=sin⁻¹(3/5), both acute. Then sin α=5/13, cos β=4/5.
- sin(α+β)=(5·4+12·3)/65=56/65; cos(α+β)=(12·4−5·3)/65=33/65>0.
- Since 0<α+β<π and its cosine is positive, α+β<π/2. Thus the sine inverse returns α+β.
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Prove cos⁻¹(4/5)+cos⁻¹(12/13)=cos⁻¹(33/65).
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