11 Establish each identity Start from left side of the equat

11) Establish each identity. Start from left side of the equation (LHS), and show that it\'s equal to right hand side (RH a) (sin x + cos x)^2 = 1 + sin 2x b) sin^2x + cos^2x=cos^2 x I c) sin Theta(cot Theta +tan Theta)=sec Theta d) csc Theta- cot Theta=sin Theta/a+cos Theta

Solution

1.

LHS = (sin x + cos x)^2

= sin^2 x + cos^2 x + 2 sin x cos x

= 1 + sin 2x

= RHS

2.

LHS = sin^2 x + cos 2x

= sin^2 x + cos^2 x - sin^2 x

= cos^2 x

=RHS

3.

LHS = sin x (cot x + tan x)

= sinx (1/tanx + tanx)

=sinx / tan x * [1 + tan^2 x]

= sinx * cosx / sin x * sec^2 x

= cos x / cos^2 x

= 1/cos x

= sec x

4.

LHS =

=cosecx - cotx

= 1/sin x - cosx/sinx

= (1-cos x ) / sin x

=[(1-cos x ) * (1+cos x)] / [ sinx (1+cosx)]

= [1-cos^2 x] / [sin x *(1+cos x)]

= sin^2 x / [sin x *(1+cos x)]

=sinx/(1+cos x)

=RHS

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