Question

Prove the trigonometric identity:
(cosecA – sinA)(secA – cosA) = 1/(tanA + cotA)

Concepts Used

  • Reciprocal identities: cosecA = 1/sinA, secA = 1/cosA
  • Ratio identities: tanA = sinA/cosA, cotA = cosA/sinA
  • Pythagorean identity: sin²A + cos²A = 1

Step-by-Step Proof

Step 1: Simplify LHS

LHS = (cosecA – sinA)(secA – cosA)

Convert to sin and cos:

= (1/sinA – sinA)(1/cosA – cosA)

Step 2: Combine fractions

= ((1 – sin²A)/sinA)((1 – cos²A)/cosA)

Step 3: Apply identities

1 – sin²A = cos²A, 1 – cos²A = sin²A

= (cos²A/sinA)(sin²A/cosA)

= sinA·cosA

Step 4: Simplify RHS

RHS = 1/(tanA + cotA)

= 1/(sinA/cosA + cosA/sinA)

Step 5: Common denominator

= 1/((sin²A + cos²A)/(sinA·cosA))

Step 6: Apply Pythagorean identity

sin²A + cos²A = 1

= 1/(1/(sinA·cosA))

= sinA·cosA

✓ Conclusion

LHS = RHS = sinA·cosA ✅ Hence proved.

Numerical Verification

Take A = 30°:

LHS = sin30·cos30 = (1/2)(√3/2) = √3/4

RHS = 1/(tan30 + cot30) = 1/(1/√3 + √3) = 1/(4/√3) = √3/4 ✅

Video Solution

Practice Questions

  1. Prove: (1 + cot²A) / (1 + tan²A) = cot²A (Answer: LHS = cosec²A/sec²A = cot²A)
  2. Prove: (sinA – cosecA)² + (cosA – secA)² = tan²A + cot²A – 1 (Answer: Expand and simplify using identities)

Also Appears In

  • CBSE Class 10 Board Exams (3-4 marks)
  • ICSE Class 10 Mathematics
  • State Board Exams (Maharashtra, UP, Karnataka)