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Introduction

Prove that: c...

Question

Prove that: $cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) = - \sqrt{2} sin x$

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Solution

$cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) = - \sqrt{2} sin x$
Let $A = \frac{3 π}{4} + x, B = \frac{3 π}{4} - x$

$\Rightarrow L . H . S = - 2 sin ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{(\frac{3 π}{4} + x) + (\frac{3 π}{4} - x)}{2} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ sin ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{(\frac{3 π}{4} + x) - (\frac{3 π}{4} - x)}{2} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$
$= - 2 sin (\frac{3 π}{4}) sin (x) = - 2 sin (π - \frac{π}{4}) sin (x) = - 2 sin \frac{π}{4} \cdot sin x = - \sqrt{2} sin x$
$∴ L . H . S . = R . H . S .$
Hence proved.

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Similar questions

Q. Prove the following:

c o s (\frac{3 π}{4} + x) - c o s (\frac{3 π}{4} - x) = - \sqrt{2} s i n x

Prove that

$(i) cos (\frac{π}{4} + x) + cos (\frac{π}{4} - x) = \sqrt{2} cos x$

(ii) $cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) = - \sqrt{2} sin x$

(cos (\frac{π}{4} + x) + cos (\frac{π}{4} - x) - \sqrt{2} cos x)

(cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) + \sqrt{2} sin x)

cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) = \sqrt{2} . sin x

cos (\frac{3 π}{4} + x) - cos (\frac{3 π}{4} - x) = - \sqrt{m} sin x

m

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