Prove the following-cos3 -4-x-cos3 -4-x-2sin x

We have L.H.S. = cos (3π/4+x) cos (3π/4 x) = 2 sin 3π/4sin x [∵ cos(A+B) cos (A B) = 2 sin A sin B] = 2 sin ( π π/4) sin x = 2 sin π/4 sin x [∵ sin (π ...

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Byju's Answer

Standard XI

Mathematics

Trigonometric Ratios for Sum of Two Angles

Prove the fol...

Question

Prove the following:

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

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Solution

We have

L.H.S. = 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 $(π - θ)$ = sin $θ$ ]

=- $2 \times \frac{1}{\sqrt{2}}$ sin x

=- $\sqrt{2}$ sin x = R.H.S.

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Similar questions

Q. Prove that:

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

Prove that

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

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

Q. Find

(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

Prove the following:

$c o s (\frac{3 π}{2} + x) c o s (2 π + x)$ $[c o t (\frac{3 π}{2} + x) + c o t (2 π + x)]$
= 1

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Prove the following: cos(3π4+x)−cos(3π4−x)=−√2sin x

We have L.H.S. = cos (3π4+x) - cos (3π4−x) =-2 sin 3π4sin x [∵ cos(A+B)- cos (A-B) =-2 sin A sin B] =- 2 sin (π−π4) sin x =- 2 sin π4 sin x [∵ sin (π−θ) = sin θ] =- 2×1√2 sin x =- √2 sin x = R.H.S.

Prove the following:

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