Question

Tan2 theta - sin2 theta= sin2 theta × tan2 theta​

Answer 1

Hi there!

We can begin by solving the left side:

tan²Θ - sin²Θ =

Rewrite tan:

$\frac{sin^2\theta}{cos^2\theta} - sin^2\theta =$

Multiply sin²Θ by cos²Θ to get a common denominator:

$\frac{sin^2\theta}{cos^2\theta} - \frac{sin^2\theta * cos^2\theta}{cos^2\theta}=$

Subtract:

$\frac{sin^2\theta- sin^2\theta * cos^2\theta}{cos^2\theta} =$

Factor out sin²Ф from the numerator:

$\frac{sin^2\theta(1 - cos^2\theta)}{cos^2\theta} =$

Rewrite 1 - cos²Ф as sin²Ф (Pythagorean identity)

$\frac{sin^2\theta(sin^2\theta)}{cos^2\theta} =$

Simplify:

$\frac{sin^4\theta}{cos^2\theta} =$

Split into sin and tan:

$sin^2\theta * \frac{sin^2\theta}{cos^2\theta} =$

Rewrite:

$sin^2\theta * tan^2\theta$