Sec thita -1 ÷ sec thita +1 = (sin thita ÷ 1+ costhita) ^ 2?

Svar:

Se venligst beviset nedenfor

Vi behøver

# Sectheta = 1 / costheta #

# Synd ^ 2-theta + cos ^ 2-theta = 1 #

Derfor

# LHS = (sectheta-1) / (sectheta + 1) #

# = (1 / costheta-1) / (1 / costheta + 1) #

# = (1-costheta) / (1 + costheta) #

# = ((1-costheta) (1 + costheta)) / ((1 + costheta) (1 + costheta)) #

# = (1-cos ^ 2-theta) / (1 + costheta) ^ 2 #

# Synd ^ 2-theta / (1 + costheta) ^ 2 #

# = (Sintheta / (1 + costheta)) ^ 2 #

# = RHS #

# QED #

# LHS = (secx-1) / (secx + 1) #

# = (1 / cosx-1) / (1 / cosx + 1) #

# = (1-cosx) / (1 + cosx) * (1 + cosx) / (1 + cosx) #

# = (1-cos ^ 2x) / (1 + cosx) ^ 2 = sin ^ 2x / (1 + cosx) ^ 2 = (sinx / (1 + cosx)) ^ 2 = RHS #

Forklaring nedenfor

# (Secx-1) / (secx + 1) #

=# ((Secx-1) * (secx + 1)) / (secx + 1) ^ 2 #

=# ((Secx) ^ 2-1) / (secx + 1) ^ 2 #

=# (Tanx) ^ 2 / (secx + 1) ^ 2 #

=# (Sinx / cosx) ^ 2 / (1 / cosx + 1) ^ 2 #

=# ((Sinx) ^ 2 / (cosx) ^ 2) / ((1 + cosx) ^ 2 / (cosx) ^ 2) #

=# (Sinx) ^ 2 // (1 + cosx) ^ 2 #

=# (Sinx / (1 + cosx)) ^ 2 #