Use the power reducing formulas to find the trig function. tan^2 157.5 degrees

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lemjay | High School Teacher | (Level 3) Senior Educator

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`tan^2 157.5^o`

The power reduction formula of square of tangent is:

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

Applying this, the given trigonometric function becomes:

`tan^2 157.5^o `

`= (1-cos(2*157.5^o))/(1+cos(2*157.5^o))`

`=(1-cos315^o)/(1+cos315^o)`

Then, plug-in the value of cos 315^o.

`=(1-sqrt2/2)/(1+sqrt2/2)`

And, simplify it.

`=(1-sqrt2/2)/(1+sqrt2/2)* (2/1)/(2/1)`

`=(2-sqrt2)/(2+sqrt2)`

`=(2-sqrt2)/(2+sqrt2) *(2-sqrt2)/(2-sqrt2)`

`=(4-2sqrt2-2sqrt2+2)/(4-2sqrt2+2sqrt2-2)`

`=(6-4sqrt2)/2`

`=3-2sqrt2`

Therefore,  `tan^2 157.5^o=3-2sqrt2` .

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