please help with these two, thank you so much!

Question

a)Simplify and write the trigonometric expression in terms of sine and cosine:

F(x)= ?

b) If ^22ab

 then the positive power a= 

the positive power 

 

Answer 1

a) I am assuming that f(x) and F(x) are meant to be the same thing; otherwise I would not understand the question.

-1/f(x) = cot(-x)cos(-x) + sin(-x) // given

= cos(-x)cos(-x)/sin(-x) + sin(-x) // after cot = cos/sin

= cos²(-x)/sin(-x) + sin(-x) // after simplification

= (cos²(-x) + sin²(-x))/sin(-x) // after bringing both terms to the common denominator sin(-x)

= 1/sin(-x) // after applying sin² + cos² = 1

f(x) = -sin(-x) // after inverting both sides of the equation

= sin(x) // because sin is symmetrical around the origin

b) tan²t - sin²t // given

= sin²t/cos²t - sin²t // after tan = sin/cos

= sin²t (1/cos²t - 1) // after factoring out sin²t from both terms

= sin²t (1-cos²t)/cos²t // after bringing to common denominator cos²t

= sin²t sin²t / cos²t // after using sin² + con² = 1

= sin⁴t / cos²t // after combining

Therefore the identity can be satisfied with a = 4, b = 2