I need some help please Write the word or phrase that beset

I need some help please;

Write the word or phrase that beset completes each statement or answers the question. Verify that each equation is an identity. 1 + cos t/1 - cos t - 1 - cos t/1 + cos t = 4 cot t csc t sec theta - 1/tan theta = tan theta/sec theta + 1 tan x + cot x/tan x - cot x = 1/sin^2x - cps^2x sec beta + tan beta = cos beta/1 - sin beta cot x sec^4 x = cot x + 2 tan x + tan^3 x

Solution

I will answer 2 question post other three separately

1).

L.H.S= (1+cost)/(1-cost) - (1-cost)/(1+cost)

= [ (1+cost) (1+cost) - (1-cost).(1-cost) ] /(1-cost) (1+cost)

= [ (1+cost)^2 + (1-cost)^2 ] / (1-cos^2t)

but sin^2t = 1 -cos^2t

= [ 1+cos^2t +2cost -1 -cos^2t +2cost] /sin^2t

= [4cost/sint ] /sint

= 4cot(t) . csc(t)

L.H.S=R.H.S

so hence proved

2).

L.H.S = (sec -1 )/tan

now divide and multiply with (sec +1)

L.H.S = [ (sec-1) (sec +1) ] / tan(sec +1)

= [ sec^2 -1 ] /tan(sec+1)

but sec^2 - tan^2 =1

= tan^2/tan(sec+1)

=tan/(sec+1)

L.H.S=R.H.S

hence proved