1. Prove that ( 1 - cos2 x ) cosec2 x = 1 .
2. Prove that (tan x sin x ) + cos x = sec x .
3. Prove that Tan4 x + Tan2 x = sec4 x - sec2 x .
4. prove that Cot x + Tan x = sec x cosec x.
5. Prove that (1 + sin x) (1 - sin x) = (sec x - Tan x)2 .
1. Prove that (1- cos2 x) cosec 2 x =1.
1 – cos2x = sin2x
= Sin2 x cosec 2 x
Cosec 2 x= 1/sin2 x
= sin2 x . 1/sin2 x
=1 Proved
2. Prove that (tan x sin x ) + cos x = sec x .
Tan x =sin x/cos x
= (Sin x/co x)sin x + cos x
= Sin2 x/co x + cos x
= (sin2 x + cos2 x)/cos x
But sin2 x +cos2 x =1
Then; 1/cos x =sec x hence proved
3. Prove that Tan 4 x + Tan2 x = sec4 x - sec2 x
Tan4 x +Tan 2 x=Tan2x(Tan2 x + 1)
But Tan2 x = sin2 x/cos2 x
=sin2 x/cos2 x(sin2 x/cos2 x + 1)
=sin2 x/cos2 x {( sin2 x+cos2 x)/cos2 x}
=sin2 x/cos2 x .1/cos2 x
=sin2 x/cos4 x
Sin2x= 1- cos 2 x
= 1-cos2 x/cos4 x
=1/cos 4 x – cos2 x/cos4 x
=1/cos4x - 1/cos 2 x where 1/cos x = sec x
Sec4 x – sec2 x proved.
4.Prove that Cot x + Tan x = sec x cosec x.
Cot x + Tan x =cos x/ sin x + sin x/cos x
= (sin2 x + cos2 x)/(sin x cos x)
= 1/ (sin x cos x) =1/sin x .1/cos x
Where 1/sin x=cosec x
1/cos x=sec x
1/cos x .1/sin x =sec x cosec x Proved
5. Prove that (1 + sin x)/(1 - sin x) = (sec x + Tan x)2 .
( 1 + sin x) / (1 – sin x) = (1 + sin x) (1 + sin x) / (1 – sin x) (1 + sin x)
= (1 + sin x) 2/(1 – sin2 x)
= (1 + sin x)2/cos2x
= {(1 + sin x)/cos x} 2
= (1/cos x + sin x/cos x)2
= (sec x + tan x)2 proved
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