a) Solve 1 - cos(theta)/tan(theta) + sec (theta)= 0 for all values of (theta) b) explain why cos x = cos (-x)   

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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a) If you need to solve `(1 - cos theta)/(tan theta + sec theta) = 0` , then, you need to substitute `sin theta/cos theta`  for `tan theta`  and `1/cos theta ` for `sec theta ` such that:

`(1 - cos theta)/((sin theta+1) /(cos theta)) = 0`

`cos theta (1 - cos theta)/(sin theta+1) = 0`

`cos theta (1 - cos theta) = 0`

`cos theta = 0 =gt theta =+-(pi/2) + 2n*pi`

`1 - cos theta = 0`

`cos theta = 1 =gt theta = 2n*pi`

Hence, the general solutions to equation are`theta = 2n*pi ` and `theta =+-(pi/2) + 2n*pi.`

b) You need to remember that the cosine function is even, hence, `cos x = cos(-x).`

You may prove this using the identity `cos (-x) = cos(0 - x) = cos 0*cos x + sin 0*sin x` You need to substitute 1 for cos 0 and 0 for sin 0 such that:

`cos (-x) = 1*cos x + 0*sin x`

`cos (-x) = cos x`

Thus, the last line proves that `cos (-x) = cos x.`

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