verify Identities 1 sec4theta tan4theta sec2thetatan2theta

verify Identities: 1. sec^4theta- tan^4theta / sec^2theta+tan^2theta = sec^2theta - tan^2theta

2, costheta + 1 / tan^(2)theta )=(cos\\theta )/(sec\\theta -1)

Solution

1) We have given (sec⁴(theta)- tan⁴(theta)) / (sec²(theta)+tan²(theta)) = sec²(theta) - tan²(theta)

take LHS=(sec⁴(theta)- tan⁴(theta)) / (sec²(theta)+tan²(theta))

=((sec²(theta))²- (tan²(theta))²) / (sec²(theta)+tan²(theta))

use the formula a^2-b^2=(a+b)*(a-b) as a=sec²(theta) and b=tan²(theta)

=((sec²(theta))- (tan²(theta))*(sec²(theta))+(tan²(theta))) / (sec²(theta)+tan²(theta))

=sec²(theta))- (tan²(theta)

=RHS

(sec⁴(theta)- tan⁴(theta)) / (sec²(theta)+tan²(theta)) = sec²(theta) - tan²(theta)

the given Idenitity is equal

2) We have given (cos(theta)+1)/tan²(theta) =cos(theta)/(sec(theta)-1)

take LHS=(cos(theta)+1)/tan²(theta)

=(1/sec(theta) +1)/(sec²(theta)-1) since cos(theta)=1/sec(theta) and tan²(theta)=sec²(theta)-1

=[(1+sec(theta))/sec(theta)]/[(sec(theta)-1)*(sec(theta)+1)]

=[(sec(theta)+1)/sec(theta)] *[1/(sec(theta)-1)*(sec(theta)+1)]

=[1/sec(theta)]*[1/(sec(theta)-1)]

=cos(theta)/(sec(theta)-1) since cos(theta)=1/sec(theta)

=RHS

Therefore (cos(theta)+1)/tan²(theta) =cos(theta)/(sec(theta)-1)

the given identity is true