How do you prove (1-sin^2theta)(1+cot^2theta)=cot^2theta?

3 Answers
Aug 4, 2018

Please see below.

Explanation:

We know that ,

(1)cos^2x+sin^2x=1
(2)csc^2x-cot^2x=1
(3)cscx=1/sinx
(4)cosx/sinx=cotx

Using (1) and (2):

LHS=(1-sin^2theta)(1+cot^2theta)

LHS=cos^2thetacsc^2thetatoApply(3)

LHS=cos^2theta*1/sin^2theta

LHS=cos^2theta/sin^2thetatoApply(4)

LHS=cot^2theta

LHS=RHS

Aug 4, 2018

"see explanation"

Explanation:

"using the "color(blue)"trigonometric identity"

•color(white)(x)cottheta=costheta/sintheta

"consider the left side"

(1-sin^2theta)(1+cos^2theta/sin^2theta)

"expand the factors"

=1+cot^2theta-sin^2theta-cos^2theta

=1+cot^2theta-(sin^2theta+cos^2theta)

=1+cot^2theta-1larrsin^2theta+cos^2theta=1

=cot^2theta=" right side "rArr" verified"

Aug 5, 2018

As proved below.

Explanation:

![https://in.pinterest.com/pin/359021401516045145/](useruploads.socratic.org)

sin^2 theta + cos^2 theta = 1

1 - sin^2 theta = cos^2 theta

1 + cot^2 theta = csc^2 theta

csc theta = 1/ sin theta

(1 - sin^2 theta) (1 + cot^2 theta) = cos^2 theta * csc^2 theta

=> cos^2 theta * 1 / sin^2 theta

=> cos^2 theta / sin^2 theta = cot ^2 theta = R H S