Question 15 of 18intermediate🔧 ApplyShort Answer2 marks

Express the trigonometric ratios sin⁡A\sin A, sec⁡A\sec A and tan⁡A\tan A in terms of cot⁡A\cot A.

Correct Answer

We know that cosec²A = 1 + cot²A.

Since cosec A = 1/sin A, we have: 1/sin²A = 1 + cot²A Therefore, sin A = 1/1+cot2A\sqrt{1 + cot²A}

Since cot A = 1/tan A, we get: tan A = 1/cot A

Using the identity sec²A = 1 + tan²A, we have: sec²A = 1 + (1/cot A)² = 1 + 1/cot²A = (cot²A + 1)/cot²A Therefore, sec A = 1+cot2A\sqrt{1 + cot²A}/cot A

Exercise: EXERCISE 8.3 | Q: 1 | (Page Page 19)
For More Understanding

Explanation

This question is from Exercise 8.3, Question 1 of NCERT Class 10 Mathematics. The solution requires using the trigonometric identities provided in the context: cosec²A = 1 + cot²A and sec²A = 1 + tan²A. Students need to express sin A, sec A, and tan A in terms of cot A by manipulating these identities. The reciprocal relationships (cot A = 1/tan A, cosec A = 1/sin A, sec A = 1/cos A) are also essential for deriving the final expressions.

Solution Steps

  1. Step 1: Use the identity cosec²A = 1 + cot²A to find sin A in terms of cot A

  2. Step 2: Since cosec A = 1/sin A, substitute to get 1/sin²A = 1 + cot²A, therefore sin A = 1/1+cot2A\sqrt{1 + cot²A}

  3. Step 3: Use the reciprocal relationship cot A = 1/tan A to get tan A = 1/cot A

  4. Step 4: Use the identity sec²A = 1 + tan²A and substitute tan A = 1/cot A

  5. Step 5: Simplify to get sec A = 1+cot2A\sqrt{1 + cot²A}/cot A

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