Beginning of Test 4 Lectures:4.4 and 5.1BPopper 13 todayQuiz 10 next weekHomework 23 AND 24 due Saturday and Sunday respectively.1First2 2( )( )x y x y x y- = + -The Difference of Two Squares.Now let’s look at the Pythagorean Identity:2 2sin cos 1x x+ =With a little algebra note that:2 2cos 1 sin (1 sin )(1 sin )x x x x= - = + -Similarly2 2sin 1 cos (1 cos )(1 cos )x x x x= - = + -Example:Graphcostan1 sinxxx--2Continuing with the Pythagorean Identity2 2sin cos 1x x+ =divide both sides by 2sin x2 22 2 22 2sin cos 1sin sin sin1 cot cscx xx x xx x+ =+ =Which says that2 2cot csc 1 (csc 1)(csc 1)x x x x= - = + -Similarly:2 2sin cos 1x x+ =divide both sides by 2cos x2 22 2 22 22 2sin cos 1cos cos costan 1 sectan sec 1 (sec 1)(sec 1)x xx x xx xx x x x+ =+ == - = + -3Popper 13 Question 1 Quiz 10, Question 24Example:cos 1 sin1 sin cosx xx x+++Quiz 10, Question 12Example:2cos11 sinxx--HW 235Other items to remember when simplifying:Even and Odd rulescos( ) cos( ) sin( ) sin( )x x x x- = - =-Simplify:tan( )sin( )tt-HW 24cos(x)cot(x) + sin(x) HW 246Coterminal anglesSimplify:sec( 12 ) csc( 400 )1 tan( 7 )t ttp pp+ + -+ +Popper 13, Question 27Popper 13, Question 38These go better with practice. Now this is part of a larger picture: Verify the Identity.So, here’s the instructions:1. box off one side of the equation. Never, ever touch it.2. convert all expressions to sines and cosines…simplify!3. apply identities and use algebra, especially factoring4. go for maximum partial credit not perfection. but remember: perfection can happen!What am I talking about? Verifying trig identities. Establishing that a complicated expression is the same as a simpler one. Here’s one:Prove that xtanxsecxsin1xcospick one side and box it off…pick the simpler side to box off.convert all expressions to sines and cosinesapply identities and algebra (make it happen)9Now pick the other side and do it again: xtanxsecxsin1xcosProve:xsin2xcotxsinxcotxcosWhat you’ll be getting in the homework and quizzes is one side with the other side in a list of choices.10More work:Simplify (1  sin x)(sec x + tan x)Popper 13, Question 411Example: Simplify2 2(sin cos ) (sin cos )x x x x+ + -Simplify21 (sin cos )x x- +12Popper 13, Question 5 Simplify2tan cotcostan cotx xxx x-++13Simplify1 sec1 secxx+-Popper 13, Question 614Popper 13, Question 7Popper 13, Question 815Popper 13, Question 9Popper 13, Question