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Question: If \(\sin ^ { 4 } x + \cos ^ { 4 } y + 2 = 4 \sin x \cos y\) and <img src="https://cdn.pureessence.t...

If sin⁡4x+cos⁡4y+2=4sin⁡xcos⁡y\sin ^ { 4 } x + \cos ^ { 4 } y + 2 = 4 \sin x \cos y and

then sinx + cosy is equal to

A

-2

B

0

C

2

D

None of these

Answer

2

Explanation

Solution

The given equation can be written as

sin⁡4x+cos⁡4y+2−4sin⁡xcos⁡y=0\sin ^ { 4 } x + \cos ^ { 4 } y + 2 - 4 \sin x \cos y = 0 ⇒ (sin⁡2x−1)2+(cos⁡2y−1)2+2sin⁡2x+2cos⁡2y−4sin⁡xcos⁡y=0\left( \sin ^ { 2 } x - 1 \right) ^ { 2 } + \left( \cos ^ { 2 } y - 1 \right) ^ { 2 } + 2 \sin ^ { 2 } x + 2 \cos ^ { 2 } y - 4 \sin x \cos y = 0⇒ (sin⁡2x−1)2+(cos⁡2y−1)2+2(sin⁡x−cos⁡y)2=0\left( \sin ^ { 2 } x - 1 \right) ^ { 2 } + \left( \cos ^ { 2 } y - 1 \right) ^ { 2 } + 2 ( \sin x - \cos y ) ^ { 2 } = 0 which is true if , and sin⁡x=cos⁡y\sin x = \cos yas we get sinx=cosy=1

⇒ sin⁡x+cos⁡y=2\sin x + \cos y = 2