Please solve the problem 2 only ab and c continueing from th

Please solve the problem 2 only (a,b and c) continueing from the problem 1 linked below.

Problem 1:

https://www.chegg.com/homework-help/questions-and-answers/1a-let-0-arbitrary-angle-verify-sin0-cos0-1-2-sin-0-cos0-may-assume-pythagorean-identities-q22359065

Problem 2:

2. Let a and B be arbitrary angles. a. Use the result of Exercise la to prove that sin a cosa 2 b. Use the facts that sin B) sin p and cos(-B) cos B to prove that sin a cosa c. Prove that sina cosa s V2.

Solution

problem 1

a) let theta be an angle in right angled triangle

sin theta = opposite / hypotenuse = o/h

cos theta = adjacent / hypotenuse = a/h

proving

(sin theta + cos theta )^2 = 1 + 2 sin theta cos theta

( o/h + a/h ) ^2

= (o+a)^2 / h^2

= (o^2 + a^2 + 2oa )/h^2

= (o^2/h^2) + (a^2 / h^2 ) + 2 oa / h^2

= sin^2 theta + cos^2 theta + 2 sin theta cos theta

= 1 + 2 sin theta cos theta

which is right hand side

proved !

b) sin gama = o/h

cos gama = a / h

o/h + a / h = sqrt 2

(o+a) / h = sqrt 2

o+ a = sqrt 2

h = 1

o^2 + a^2 = 1

o = sqrt 2 - a

(sqrt 2 - a )^2 + a^2 = 1

2 + a^2 - 2a sqrt 2 + a^2 = 1

2a^2 - 2a sqrt 2 = -1

2a^2 + 1 = 2a sqrt 2

a = 1 / sqrt 2

o = sqrt 2 - 1/ sqrt 2

o = 1 / sqrt 2

hence , tan gamma = o / a

tan gamma = 1

gamma = 45 degrees