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welshfella
-85 degrees in is the 4th quadrant so the cos and sec  will be positive
sec is also positive in first quadrant so sec(85)  will be equal to sec(-85)
Sec is is negative in 3rd quadrant so -sec(-95) will be same also.

sec 275 is in 3rd quadrant  so will be negative  so not equal
also sec(-95) will be same angle as sec 275

the last 2 options are not equal to sec(-85)

Answer:  The correct option is (D) [tex]\sec(-95^\circ).[/tex]

Step-by-step explanation:  We are given to select the correct option that is not equal to [tex]\sec(-85^\circ).[/tex]

Option (A) :  [tex]\sec(85^\circ).[/tex]

Since secant of any angle is an even function, so

[tex]\sec(-\theta)=\sec\theta.[/tex]

Therefore,

[tex]\sec(-85^\circ)=\sec85^\circ.[/tex]

So, this option is not correct.

Option (B) :  [tex]-\sec(-95^\circ).[/tex]

We have

[tex]-\sec(-95^\circ)=-\sec95^\circ=-sec(2\times90^\circ-85^\circ)=-(-\sec85^\circ)=\sec85^\circ=\sec(-85^\circ).[/tex]

So, this option is also not correct.

Option (C) :  [tex]\sec(275^\circ).[/tex]

We have

[tex]\sec275^\circ=\sec(4\times90^\circ-85^\circ)=\sec85^\circ= \sec(-85^\circ).[/tex]

So, this option is incorrect.

Option (D) :  [tex]\sec(-95^\circ).[/tex]

We have

[tex]\sec(-95^\circ)=\sec95^\circ=\sec(2\times90^\circ-85^\circ)=-\sec85^\circ\neq \sec(-85^\circ).[/tex]

So, this option is correct.

Thus, (D) is the correct option.

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