Question

Tan2 theta - sin2 theta= sin2 theta × tan2 theta
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Answer 1

Hi there!

We can begin by solving the left side:

tan²Θ - sin²Θ =

Rewrite tan:

`(sin^2\theta)/(cos^2\theta) - sin^2\theta =`

Multiply sin²Θ by cos²Θ to get a common denominator:

`(sin^2\theta)/(cos^2\theta) - (sin^2\theta * cos^2\theta)/(cos^2\theta)=`

Subtract:

`(sin^2\theta- sin^2\theta * cos^2\theta)/(cos^2\theta) =`

Factor out sin²Ф from the numerator:

`(sin^2\theta(1 - cos^2\theta))/(cos^2\theta) =`

Rewrite 1 - cos²Ф as sin²Ф (Pythagorean identity)

`(sin^2\theta(sin^2\theta))/(cos^2\theta) =`

Simplify:

`(sin^4\theta)/(cos^2\theta) =`

Split into sin and tan:

`sin^2\theta * (sin^2\theta)/(cos^2\theta) =`

Rewrite:

`sin^2\theta * tan^2\theta`