Prove thati cos-4-x-cos-4-x-2cos xii cos3 -4-x-cos3 -4-x-2sin x

(i)LHS = cos ( π/4 + x ) + cos ( π/4 x ) = 2 cos π/4 cos x [∵cos (A + B) + cos (A B) = 2 cosA cos B] = ( 2 ×1/√(2)cos x ) = √(2) cos x = RHS. (ii) ...

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Byju's Answer

Standard XII

Mathematics

Sin(A+B)Sin(A-B)

Prove thati c...

Question

Prove that

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

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

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Solution

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

$= 2 cos \frac{π}{4} cos x [∵ cos (A + B) + cos (A - B) = 2 c o s A cos B]$

$= (2 \times \frac{1}{\sqrt{2}} cos x) = \sqrt{2} cos x = RHS.$

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

= $- 2 sin \frac{3 π}{4} sin x$

$[∵ cos (A + B) - cos (A - B) = - 2 sin A sin B]$

= $- 2 sin (π - \frac{π}{4}) sin x$

$= - 2 sin \frac{π}{4} sin x [∵ sin (π - \frac{π}{4}) = sin \frac{π}{4}]$

= $(- 2 \times \frac{1}{\sqrt{2}}) sin x = - \sqrt{2} s i n x = R H S .$

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Prove that (i) cos(π4+x)+ cos(π4−x) = √2 cos x (ii) cos(3π4+x)− cos(3π4−x) =− √2 sin x

Prove that

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

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