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If ^{-1} [{ }] - ^{-1} [{ }] = x, then x is equal to

Question

If $cot^{- 1} [cos α] - tan^{- 1} [cos α] = x$ , then $sin x$ is equal to

tan^{2} (\frac{α}{2})

cot^{2} (\frac{α}{2})

tan α

cot \frac{α}{2}

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cot^{- 1} [cos α] - tan^{- 1} [cos α] = x

, then

sin x

is equal to

6 mins ago

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6 mins ago

Text solutionVerified

Given equation:

co t^{- 1} (cos α) - t an^{- 1} (cos α) = x

l e t co t^{- 1} (cos α) = P

⟹ co tP = cos α

⟹ s in P = \frac{1}{1 + cos α}

Also

t an^{- 1} (cos α) = Q

⟹ t an Q = cos α

⟹ s in Q = \frac{cos α}{1 + cos α}

Now the given equation can be written as

x = P - Q

s in x = s in (P - Q)

s in x = s in P cos Q - cos P s in Q

s in x = (\frac{1}{1 + cos α}) (\frac{1}{1 + cos α}) - (\frac{cos α}{1 + cos α}) (\frac{cos α}{1 + cos α})

s in x = \frac{1 - cos α}{1 + cos α} = \frac{2 s in ^{2} \frac{α}{2}}{2 cos ^{2} \frac{α}{2}} s in x = t an^{2} \frac{α}{2}

Hence option (a) is correct.

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Question Text	If $cot^{- 1} [cos α] - tan^{- 1} [cos α] = x$ , then $sin x$ is equal to
Answer Type	Text solution:1
Upvotes	150