Prove that
`(sintheta"cosec"thetatanthetacottheta)/(sin^(2)theta+cos^(2)theta)=1`

Prove that : sintheta/cos(3theta)+sin(3theta)/cos(9theta)+sin(9theta)/cos(27theta)=1/2(tan(27theta)-tan(theta))

Prove that sin^(3)theta+cos^(3)theta=(sintheta+costheta)(1-sin thetacostheta) .

Prove that : (sintheta-costheta)/(sintheta+costheta)+(sintheta+costheta)/(sintheta-costheta)=(2)/(2sin^(2)theta-1)

If int(sintheta-costheta)/((sintheta+costheta)sqrt(sinthetacostheta+sin^(2)thetacos^(2)theta))d theta = "cosec"^(_1)(f(theta))+C , then

Prove that: a) (sin2theta)/(1+cos2theta) = tantheta b) (1+sin2theta+cos2theta)/(1+sin2theta-cos2theta)=cottheta

Prove that sin(2theta)/(1-cos2theta)=cot theta

Prove that cos^2 theta cosec theta+sin theta=cosec theta

Prove that cos^(2)theta("cosec"^(2)theta-cot^(2)theta)=cos^(2)theta .

Prove that, sin^2 theta+ cosec^2 theta>=2

From algebra we know that if ax^(2) +bx + c=0 , a( ne 0), b, c int R has roots alpha and beta then alpha + beta=-b/a and alpha beta = c/a . Trignometric functions sin theta and cos theta, tan theta and sec theta, " cosec " theta and cot theta obey sin^(2)theta + cos^(2)theta =1 . A linear relation in sin theta and cos theta, sec theta and tan theta or " cosec "theta and cos theta can be transformed into a quadratic equation in, say, sin theta, tan theta or cot theta respectively. And then one can apply sum and product of roots to find the desired result. Let a cos theta, b sintheta=c have two roots theta_(1) and theta_(2) . theta_(1) ne theta_(2) . The vlaue of cos(theta_(1) + theta_(2)) is a and b not being simultaneously zero)