Question

if root 3 tan theta = 3 sin theta, find the value of sin² theta - cos² theta

Answer 1

3tantheta=9sin^2theta
sintheta/costheta=3sin^2theta
1/3=sinthetacostheta
1/3=2/2sinthetacostheta
1/3=1/2sin2theta
2/3=sin2theta ----------(1)

cos^2theta-sin^2theta=cos2theta

So

sin^2theta-cos^2theta= --cos2theta--------(2)

From eq 1

sin2theta=2/3

we know cos^2(2theta)= 1--sin^2(2theta)

cos^2(2theta)=1--4/9
cos^2(2theta)=5/9
cos(2theta)=root{(5)}/3

So from eq (2)

sin^2theta-cos^2theta= --cos2theta--------(2)

substituting:

==》
$\frac{ - \sqrt{5} }{3}$

Answer 2

i given the answer in the images