упростите sin^2b(1+ctgb)+cos^2b(1+tgb). Created by Nishinoyaa. algebra-ru

упростите sin^2b(1+ctgb)+cos^2b(1+tgb)...

Artem112

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$\sin^2b(1+\mathrm{ctg}b)+\cos^2b(1+\mathrm{tg}b)= \\\ =\sin^2b\left(1+\dfrac{\cos b}{\sin b}\right)+\cos^2b\left(1+\dfrac{\sin b}{\cos b}\right)= \\\ =\sin^2b+\sin b \cos b+\cos^2b+\sin b\cos b=1+2\sin b \cos b=1+\sin 2b$

8 votes Thanks 6

Mihail001192

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Sin²b×( 1 + ctgb ) + cos²b×( 1 + tgb)

Расскроем скобки и вспомним тригонометрические функции:
sin²b + cos²b = 1
ctgb = cosb/sinb
tgb = sinb/cosb
sin2b = 2sinb×cosb

sin²b + (sin²b×cosb)/ sinb + cos²b + (cos²b×sinb)/cosb =

sin²b + cos²b + sinb×cosb + cosb×sinb =

= 1 + 2sinb×cosb = 1 + sin2b

Ответ: 1 + sin2b

4 votes Thanks 6

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