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Question
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
Options
- \[\pm \frac{3}{4}\]
- \[\pm \frac{4}{3}\]
- \[\pm \frac{1}{3}\]
None of these
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Solution
If sin (π cos x) = cos (π sin x), then sin 2x = `underlinebb(+- 3/4)`.
Explanation:
sin (π cos x) = cos (π sin x)
∵ sin θ = cos (90 - θ)
`=> cos (pi/2 - pi cos x) = cos (pi sin x)`
`=> pi/2 - pi cos x = pi sin x`
`=> pi/2 = pi (sin x + cos x)`
`=> cancel(pi)/2 = cancel(pi) (sin x + cos x)`
`=> sin x + cos x = 1/2`
Squaring on both sides,
`=> (sin x + cos x)^2 = 1/4`
`=> sin^2x + cos^2x + 2 sin x * cos x = 1/4`
`=> 1 + 2 sin x * cos x = 1/4 ...[sin^2x + cos^2x = 1]`
`=> sin 2x = 1/4 - 1`
⇒ sin 2x = `(- 3)/4`
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