Math Problem Statement

If A+B = 90°then what’s the value of 1+ cotA/cotB = cosec²A

Solution

Given the condition A+B=90A + B = 90^\circ, we need to find the value of the expression 1+cotAcotB1 + \frac{\cot A}{\cot B}.

Step-by-Step Solution:

  1. Complementary Angle Relationship: Since A+B=90A + B = 90^\circ, B=90AB = 90^\circ - A. This implies that BB is the complementary angle of AA.

  2. Cotangent of a Complementary Angle: The cotangent of a complementary angle is: cot(90A)=tanA\cot(90^\circ - A) = \tan A Therefore, cotB=tanA\cot B = \tan A.

  3. Substitute into the Expression: The expression we want to simplify is: 1+cotAcotB1 + \frac{\cot A}{\cot B} Substitute cotB=tanA\cot B = \tan A into the expression: 1+cotAtanA1 + \frac{\cot A}{\tan A}

  4. Simplify the Expression: Notice that: cotAtanA=1tanAtanA=1tan2A=cot2A\frac{\cot A}{\tan A} = \frac{\frac{1}{\tan A}}{\tan A} = \frac{1}{\tan^2 A} = \cot^2 A So, the expression simplifies to: 1+cot2A1 + \cot^2 A

  5. Use the Pythagorean Identity: The Pythagorean identity for cotangent and cosecant is: 1+cot2A=csc2A1 + \cot^2 A = \csc^2 A

Conclusion:

The value of the expression 1+cotAcotB1 + \frac{\cot A}{\cot B} is equal to csc2A\csc^2 A.

Would you like any more details or have any other questions?

Related Questions:

  1. How can the identity 1+cot2A=csc2A1 + \cot^2 A = \csc^2 A be derived?
  2. What is the relationship between the trigonometric functions of complementary angles?
  3. How does the Pythagorean identity apply to other trigonometric functions?
  4. Can you find the value of cot2A\cot^2 A if cscA=2\csc A = 2?
  5. What are some common trigonometric identities involving sin\sin, cos\cos, and tan\tan?

Tip: Understanding complementary angles and their identities can significantly simplify solving trigonometric problems.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Complementary angles
Trigonometric identities

Formulas

Pythagorean identity: 1 + cot^2 A = csc^2 A

Theorems

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Suitable Grade Level

High school