Wie verifizieren Sie die Identität tanthetacsc ^ 2theta-tantheta = cottheta?

Antworten:

Beweis unten

Erläuterung:

#tantheta * csc ^ 2theta - tantheta #

# = sintheta / costheta * (1 / sintheta) ^ 2 - sintheta / costheta #

# = sintheta / costheta * 1 / sin ^ 2theta - sintheta / costheta #

# = 1 / (sinthetacostheta) - sintheta / costheta #

# = (1-sin ^ 2theta) / (sinthetacostheta) #

# = cos ^ 2theta / (sinthetacostheta) #

# = costheta / sintheta #

# = cottheta #

Beachten Sie, dass # sin ^ 2theta + cos ^ 2theta = 1 #, deshalb # cos ^ 2theta = 1- sin ^ 2theta #

# LHS = Tantheta * csc ^ 2theta - Tantheta #

# = Tantheta (csc ^ 2theta - 1) #

# = Tantheta (1 + Kinderbett ^ 2theta - 1) #

# = Tantheta * Kinderbett ^ 2theta #

# = cottheta = RHS #