By means of fundamental identities, express each of the following in terms of sinθ only: tanθShow complete solution and explain the answer.

Expert Answers
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You need to remember what basic trigonometric identity is:

`sin^2theta + cos^2theta = 1`

You need to remember that the tangent function is rational such that: `tan theta`  = `sin theta/cos theta`

Hence, you need to form tangent function in the basic trigonometric identity by dividing identity by `cos^2 theta` . You need to divide both sides by `cos^2 theta`  to preserve the identity.

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

`tan^2 theta + 1 = 1/(cos^2 theta)`

You need to write cos alpha in terms of sin alpha such that:

`cos^2 theta= 1 - sin^2 theta`

`tan^2 theta = 1/(1 - sin^2 theta) - 1`

Bringing the terms to a common denominator yields:

`tan^2 theta = (1 - 1 + sin^2 theta)/(1 - sin^2 theta)`

`tan^2 theta = (sin^2 theta)/(1 - sin^2 theta)`

`tan theta = +- (sin theta)/(sqrt(1 - sin^2 theta))`

Hence,writing the tangent function in terms of sin theta yields:

`tan theta = +- (sin theta)/(sqrt(1 - sin^2 theta)).`