Question

Simplify ((sin x + tan x) ^ 2 + cos^2 x - sec^2 x)/(tan x)

I know the answer is 2sin(x) but I need the work for the problem

Answer 1

Expand the numerator:

(sin(x) + tan(x))² + cos²(x) - sec²(x)

= sin²(x) + 2 sin(x) tan(x) + tan²(x) + cos²(x) - sec²(x)

Recall the Pythagorean identity:

cos²(x) + sin²(x) = 1

Multiplying both sides by 1/cos²(x) gives an equivalent form,

1 + tan²(x) = sec²(x)

so that

tan²(x) - sec²(x) = -1

Then we have

(sin(x) + tan(x))² + cos²(x) - sec²(x)

= 1 + (-1) + 2 sin(x) tan(x)

= 2 sin(x) tan(x)

and so

((sin(x) + tan(x))² + cos²(x) - sec²(x))/tan(x)

= (2 sin(x) tan(x))/tan(x)

= 2 sin(x)