11) Establish each identity. Start from left side of the equation (LH.pdf

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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 if you have further query, then ask in comment..

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Math 105 - Section4- Identities Name Use the fundamental identities to show that the left and the right hand side are equal. DO NOT MOVE TERMS OR FACTORS FROM ONE SIDE TO THE OTHER OF THE IDENT 1 + cos2 cos2 tan =sin sec 2) 3) sin2 + sin -1+cos2=sin 4) cot sin -tan cos =cos -sin sec csc tan cot 5) sin cos 6)sec + csc =-sine cos 7)(cot + tan )sec = sin cos2 Solution 1) sec^2 theta ( 1 + cos^2 theta ) = ( 1 + cos^2 theta ) / cos^2 theta manipulating left hand side sec^2 theta = ( 1 / cos^2 theta ) 1 / cos ^2 theta ( 1 + cos^2 theta ) or ( 1 + cos^2 theta ) / cos^2 theta which is right hand side proved ! 2) tan theta / sec theta = sin theta manipulating left hand side tan theta = sin theta / cos theta sec theta = 1/ cos theta so , tan theta / sec theta can be written as sin theta / cos theta / 1 / cos theta = sin theta which is right side proved ! 3) sin^2 theta + sin theta -1 + cos^2 theta = sin theta manipulating left hand side sin^2 theta = 1 - cos^2 theta so we can write left side as 1 - cos^2 theta + sin theta -1 + cos^2 theta simplifying we get sin theta on left side which is equal to right sde proved ! 4) cot theta sin theta - tan theta cos theta = cos theta - sin theta manipulating left hand side cot theta = cos theta / sin theta tan theta = sin theta / cos theta cos theta / sin theta * sin theta - sin theta / cos theta * cos theta cos theta - sin theta which is right side proved !.

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