Prove that: `("sin"(x+theta))/("sin"(x+varphi))=cos(theta-varphi)+cot(x+varphi)sin(theta-varphi)dot`

Prove that: (sin(x+theta))/(sin(x+phi))=cos(theta-phi)+cot(x+phi)sin(theta-phi)

Prove that: (sin2 theta)/(1-cos2 theta)=cot theta

Prove that: (sin(theta+varphi)-2sin theta+sin(theta-varphi))/(cos(theta+varphi)-2cos theta+cos(theta-varphi))=tan theta

(tan(theta+varphi)+tan(theta-varphi))/(1-tan(theta+varphi)tan(theta-varphi))=tan2 theta

Prove that (1+cos theta)/(sin theta)=cot[(theta)/(2)]

If sin theta+sin varphi=a and cos theta varphi+cos varphi=b then cos((theta-varphi)/(2))=+-(1)/(2)sqrt(a^(2)+b^(2))cos((theta-varphi)/(2))=+-sqrt(a^(2)-b^(2))tan((theta-varphi)/(2))=+-sqrt((4-a^(2)-b^(2))/(a^(2)+b^(2)))cos(theta-varphi)=(a^(2)+b^(2)-2)/(2)

If determinant |"cos"(theta+varphi)-"sin"(theta+varphi)cos2varphis inthetacosthetas invarphi-costhetas inthetacosvarphi| is a. positive b. independent of q c. independent of varphi d. none of these

Factors of cos4theta-cos4varphi are- a. (costheta+cosvarphi) b. \ (costheta-cosvarphi) c. ("costheta"+"sin") d. (costheta-s invarphi)

Prove that (sin theta)/((cot theta+cosec theta))-(sin theta)/((cot theta-cosec theta))=2

Prove that: 1-(sin^(2)theta)/(1+cot theta)-(cos^(2)theta)/(1+tan theta)=sin theta cos theta