Multiple choice

Find the value of (cos 60º / sin 30º) + (cos 65º. cosec 25º / tan 10º. tan 30º. tan 45º. tan 60º. tan 80º)?

  1. 1

  2. -1

  3. 0

  4. 2

  5. 3
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D Correct answer
Explanation

First term: cos 60/sin 30 = (1/2)/(1/2) = 1. Second term: cos 65 * cosec 25 = cos 65 / sin 25 = 1 (since sin 25 = cos 65). Denominator: tan 10 * tan 30 * tan 45 * tan 60 * tan 80 = (tan 10 * tan 80) * (tan 30 * tan 60) * tan 45 = 1 * 1 * 1 = 1. Total = 1 + 1/1 = 2.

AI explanation

Evaluate the given expression by first applying the complementary angle identities to the second term, such as cos 65 = sin 25 and tan 80 = cot 10. The product (sin 25)(cosec 25) simplifies to 1, and (tan 10)(cot 10) simplifies to 1. The expression becomes (cos 60 / sin 30) plus 1 / (tan 30 * tan 45 * tan 60). The terms tan 30 and tan 60 cancel out to 1, leaving (cos 60 / sin 30) plus 1 / tan 45. Since cos 60 equals sin 30 and tan 45 is 1, the calculation is 1 plus 1, which equals 2.

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