Answer:
Step-by-step explanation:
1.
cot x sec⁴ x = cot x+2 tan x +tan³x
L.H.S = cot x sec⁴x
=cot x (sec²x)²
=cot x (1+tan²x)² [ ∵ sec²x=1+tan²x]
= cot x(1+ 2 tan²x +tan⁴x)
=cot x+ 2 cot x tan²x+cot x tan⁴x
=cot x +2 tan x + tan³x [ ∵cot x tan x [tex]=\frac{ \textrm{tan x }}{\textrm{tan x}}[/tex] =1]
=R.H.S
2.
(sin x)(tan x cos x - cot x cos x)=1-2 cos²x
L.H.S =(sin x)(tan x cos x - cot x cos x)
= sin x tan x cos x - sin x cot x cos x
[tex]=\textrm{sin x cos x }\times\frac{\textrm{sin x}}{\textrm{cos x} } - \textrm{sinx}\times\frac{\textrm{cos x}}{\textrm{sin x}}\times \textrm{cos x}[/tex]
= sin²x -cos²x
=1-cos²x-cos²x
=1-2 cos²x
=R.H.S
3.
1+ sec²x sin²x =sec²x
L.H.S =1+ sec²x sin²x
=[tex]1+\frac{{sin^2x}}{cos^2x}[/tex] [[tex]\textrm{sec x}=\frac{1}{\textrm{cos x}}[/tex]]
=1+tan²x [tex][\frac{\textrm{sin x}}{\textrm{cos x}} = \textrm{tan x}][/tex]
=sec²x
=R.H.S
4.
[tex]\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}} = \textrm{2 csc x}[/tex]
L.H.S=[tex]\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}}[/tex]
[tex]=\frac{\textrm{sinx(1+cos x)+{\textrm{sinx(1-cos x)}}}}{\textrm{(1-cos x)\textrm{(1+cos x})}}[/tex]
[tex]=\frac{\textrm{sinx+sin xcos x+{\textrm{sinx-sin xcos x}}}}{{(1-cos ^2x)}}[/tex]
[tex]=\frac{\textrm{2sin x}}{sin^2 x}[/tex]
= 2 csc x
= R.H.S
5.
-tan²x + sec²x=1
L.H.S=-tan²x + sec²x
= sec²x-tan²x
=[tex]\frac{1}{cos^2x} -\frac{sin^2x}{cos^2x}[/tex]
[tex]=\frac{1- sin^2x}{cos^2x}[/tex]
[tex]=\frac{cos^2x}{cos^2x}[/tex]
=1
