If `t a n^2theta=2t a n^2varphi+1` , prove that `cos2theta+s in^2varphi=0.`

Prove that 1-cos^2theta-sin^2theta=0

If 3 tan theta tan phi=1 , then prove that 2 cos (theta+varphi)=cos (theta-varphi) .

Let theta, varphi varepsilon [0,2 pi] be such that 2cos theta (1-sin varphi) = sin ^ (2) theta ((tan theta) / (2) + (cot theta) / (2)) cos varphi -1, tan (2 pi-theta)> 0 and -1> sin theta <(sqrt (3)) / (2)

If tan theta=(1)/(2) andtan varphi =(1)/(3), then the value of theta+varphi is (pi)/(6) (b) pi(c)0(d)(pi)/(4)

If quad 2 theta+s in2 varphi=(1)/(2) and cos2 theta+cos2 varphi=(3)/(2), then cos^(2)(theta-varphi)=3/8b.5/8c.3/4d5/4

(i) If "tan""(x)/(2)=(m)/(n) ,prove that m sin x+n cos x=n. (ii) If "tan" theta=(a)/(b) , prove that a sin 2theta+b cos 2theta=b .

Prove that the product of matrices [cos^(2)theta cos theta sin theta cos theta sin theta s in^(2)theta] and [cos^(2)varphi cos varphi sin varphi cos varphi sin varphi s in^(2)varphi] is the null matrix,when theta and varphi differ by an odd multiple of (pi)/(2) .

(a+b)tan(theta-varphi)=(a-b)tan(theta+varphi), then (sin(2 theta))/(sin(2 varphi)) is equal to a.ab b.(a)/(b) c.(b)/(a) d.a^(2)b^(2)

If theta=(pi)/(2^(n)+1), prove that: 2^(n)cos theta cos2 theta cos2^(2)...*cos2^(n-1)theta=1