Answer:
Step-by-step explanation:
1)
[tex]cot x sec^4 x = cotx + 2tanx + tan^3 x[/tex]
[tex]RHS =[/tex]
[tex]=\frac{cosx}{sinx} + 2 \frac{sinx }{cosx} + \frac{sin^3x }{cos^3x}\\\\[/tex]
[tex]= \frac{cosx(cos^3x) + 2sinx(sinx \ cos^2x ) + sin^3x(sinx)}{sinx \ cos^3x}\\\\[/tex] [tex][\ taking LCM \ ][/tex]
[tex]=\frac{cos^4x + 2sin^2xcos^2x +sin^4x }{sinx cos^3x}[/tex]
[tex]= \frac{(sin^2 x + cos^2x)^2 }{sin x \ cos^3 x}[/tex] [tex][ \ a^4 + 2a^2b^2 + b^4 = (a^2 + b^2 ) ^2 \ ][/tex]
[tex]= \frac{1}{sinx \ cos^3 x}\\\\= \frac{1 \times cosx}{sinx \times cos^3x \times cosx}[/tex] [tex][ \ multiplying\ and \ dividing \ by \ cosx \ ][/tex]
[tex]= \frac{cosx}{sinx} \times \frac{1}{cos^4x}\\\\=cot x \ sec^4 x[/tex]
[tex]= LHS[/tex]
2)
[tex]sin x \ ( tanx \ cosx - cotx \ cosx) = 1 - 2cos^2x[/tex]
[tex]LHS =[/tex]
[tex]=sinx( [ \frac{sinx}{cosx} \times cosx)] - [ \frac{cosx}{sinx}\times cosx] )\\\\= sinx (sinx - \frac{cos^2x}{sinx})\\\\=sin^2x - cos^2 x\\\\=(1 - cos^2x ) -cos^2 x[/tex] [tex][ \ sin^2x = 1 -cos^2 x \ ][/tex]
[tex]= 1 -cos^2x - cos^2 x \\\\= 1 - 2cos^2x \\\\=RHS[/tex]
3)
[tex]1 + sec^2x \ sin^2x = sec^2 x[/tex]
[tex]LHS =[/tex]
[tex]= 1 +sec^2x \sin^2x \\\\= 1 + (\frac{1}{cos^2x} \times sin^2x )\\\\= 1 + \frac{sin^2 x}{cos^2x}\\\\= 1 + tan^2x \\\\= sec^2 x\\\\=RHS[/tex]
4)
[tex]\frac{sinx}{1 -cosx} + \frac{sinx}{1+cosx} = 2 \ cosec x[/tex]
[tex]LHS =[/tex]
[tex]=\frac{sinx}{1 -cosx} + \frac{sinx}{1+cosx} \\\\= \frac{sinx(1 +cosx)}{(1-cosx)(1+cosx)} + \frac{sinx(1-cox)}{(1+cosx)(1-cosx)}\\\\= \frac{sinx +sinx\ cosx}{(1 - cos^2x)} + \frac{sinx - sinx \ cosx}{1 - cos^2x}\\\\=\frac{sinx + sinx \ cosx + sinx - sinx \ cosx}{1 - cos^2x}\\\\=\frac{2sinx}{sin^2x}\\\\=\frac{2}{sinx}\\\\=2 cosec\ x\\\\=RHS[/tex]
5)
[tex]- tan^2 + sec^2 x = 1\\\\[/tex]
[tex]sin^2 x + cos^2 x = 1\\\\\frac{sin^2x }{cos^2x} + \frac{cos^2x}{cos^x} = \frac{1}{cos^2x}\\\\tan^2x + 1 = sec^2x \\\\1 = sec^2 - tan^2x \\\\-tan^2x + sec^2 x = 1[/tex]
