Can someone help me verify the identities for these please? Thanks!

enter image source here

1 Answer
Jan 3, 2018

Proofs below

Explanation:

a)
We first use the facts that sinu/cosu = tanu and cos^2u+sin^2u =1:
(1+tan^2u)(1-sin^2u) = (1+sin^2u/cos^2u)(cos^2u)
Expand:
=(cos^2u+sin^2u)
And done:
=1

b)
We need to use the cosine double angle expansions cos(alpha-beta) = cosalphacosbeta+sinalphasinbeta and cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta:
using these:
cos(alpha-beta)-cos(alpha+beta)=cosalphacosbeta+sinalphasinbeta-(cosalphacosbeta-sinalphasinbeta)
=2sinalphasinbeta

c)We use the definitions of cot and csc:
cot^2x+csc^2x=cos^2x/sin^2x+1/sin^2x
=(cos^2x+1)/sin^2x
=(1-sin^2x+1)/sin^2x
=(2-sin^2x)/sin^2x
=2csc^2x-1

d)Now we use the definition of sec
1+sec^2xsin^2x=1+sin^2x/cos^2x
=(cos^2x+sin^2x)/cos^2x
=1/cos^2x
=sec^2x