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Prove that #((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2# ?

Answer 1

Elijah Alexander

Please see below.

We use formulas (A) - #cosA=sin(90^@-A)#,

(B) - #cos^2A-sin^2A=cos2A#

(D) - #sinA+sinB=2sin((A+B)/2)cos((A-B)/2)# and

(E) - #sinA-sinB=2cos((A+B)/2)sin((A-B)/2)#

#(cos^2 33^@-cos^2 57^@)/(sin^2 10.5^@-sin^2 34.5^@)#

= #(cos^2 33^@-sin^2 (90^@-57^@))/((sin10.5^@+sin34.5^@)(sin10.5^@-sin34.5^@))# - used A

= #(cos^2 33^@-sin^2 33^@)/(-(2sin22.5^@cos12^@)(2cos22.5^@sin12^@))# - used D & E

= #(cos66^@)/(-(2sin22.5^@cos22.5^@xx2sin12^@cos12^@)# - used B

= #-(sin(90^@-66^@))/(sin45^@sin24^@)# - used A & C

= #-sin24^@/(1/sqrt2sin24^@)#

= #-sqrt2#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.