express in anglecsc (90 degrees + angle) sin (180 degrees + angle) - tan (270 degrees - angle)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to come up with the following substitution such that angle = `alpha` .

You need to remember that cosecant is reciprocal of sine function such that:

`csc (90+alpha) = 1/sin(90+alpha)`

`sin (90+alpha) = cos alpha =gt csc (90+alpha) = 1/(cos alpha)`

You need to remember that secant function is reciprocal of cosine function such that:

`1/(cos alpha) = sec alpha`

`sin (180 + angle) = sin180*cos alpha + sin alpha*cos 180 = -sin alpha`

Hence `csc (90+alpha)*sin (180 + angle) = -sin alpha/cos alpha = -tan alpha`

You need to calculate `tan(270-alpha) = sin (270-alpha)/cos (270-alpha)`

`sin (270-alpha) = sin 270*cos alpha - sin alpha cos 270 = -cos alpha`

`cos (270-alpha) = cos 270*cos alpha + sin 270*sin alpha = -sin alpha`

`tan(270-alpha) = -sin alpha/(- cos alpha) = tan alpha`

Evaluating the expression yields:

csc (90+alpha)*sin (180 + angle) - tan(270-alpha) =-tan alpha - tan alpha = -2tan alpha

Hence, expressing  csc (90+alpha)*sin (180 + angle) - tan(270-alpha)  in terms of `alpha`  yields  - 2tan alpha .

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