Verify: (sinx+cosx) ^2=1+sin (2x)

Question

Erika Wilson

Mathematics

2 January, 18:21

Verify: (sinx+cosx) ^2=1+sin (2x)

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Rory Glover · Answer 1

(sinx + cosx) ² = 1 + sin (2x)

Since (a + b) ² = (a + b) (a+b):

(sinx + cosx) (sinx + cosx) = 1 + sin (2x)

Now, multiply each member with another:

sin²x + sinxcosx + sinxcosx + cos²x = 1 + sin (2x)

⇒ sin²x + cos²x + 2sinxcosx = 1 + sin (2x)

Since sin²x + cos²x = 1 (Pithagorean identity):

1 + 2sinxcosx = 1 + sin (2x)

Since 1 + 2sinxcosx = 1 + sin (2x) (Double-angle identity):

1 + sin (2x) = 1 + sin (2x)