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

Odgovor:

Pogledajte dokaz u nastavku

Obrazloženje:

Trebamo

# Sectheta = 1 / costheta #

# Grijeh ^ 2 theta + cos ^ 2 theta = 1 #

Stoga

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

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

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

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

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

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

Odgovor:

Objašnjenje u nastavku

Obrazloženje:

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

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

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

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