Question

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

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)