Prove that (Sin theta+cosec theta)2 +(Cos theta +sec theta)2 = 7+ Tan2 theta + Cot2 theta?

Answer 1

(sin t + cosec t)^2 = sin ^2 t + cosec ^2 t + 2
(cos t + sec t)^2 = cos^2 t + sec ^2 t + 2

using (a+b)^2 = a^2 +b^2 + 2ab and sint cosec t = 1

add the 2
LHS -= sin ^2 t + cos ^2 t + (cot^2 t +1) + 2 + (tan ^2 t + 1) +2
= 1 + (cot^2 t +1) + 2 + (tan ^2 t + 1) +2
= 7 + tan ^2 t + cot^2 t
= RHS

Answer 2

LHS = (Sin θ +csc θ)^2 +(Cos θ + sec θ)^2 = (Sin θ + 1/sin θ)^2 +(Cos θ + 1/cos θ)^2 = [(1+sinθ)/sinθ]^2 + [(1+cosθ)/cosθ]^2 = (1+sinθ)^2/(sinθ)^2 + (1+cosθ)^2/(cosθ)^2 = [(cosθ)^2*(1+sinθ)^2 + (sinθ)^2*(1+cosθ)^2]/[(sinθ)^2*(cosθ)^2]
expand the numerator and use (sinθ)^2 + (cosθ)^2 = 1, we have
LHS = [1 + 5(sinθ)^2 * (cosθ)^2]/[(sinθ)^2 * (cosθ)^2] = 5 + 1/[(sinθ)^2 * (cosθ)^2] = 5 + [(sinθ)^2 + (cosθ)^2]/[(sinθ)^2 * (cosθ)^2] = 5 + (sec θ)^2 + (cscθ)^2 = 5 + 1 + (tanθ)^2 + 1 + (cotθ)^2 = 7 + (tanθ)^2 + (cotθ)^2 = RHS

Answer 3

(sinθ + cosecθ)^2 + (cosθ +secθ)2 = 7 + tan^2θ + cot^2θ
sin^2θ + 2 + cosec^2θ + cos^2θ + 2 + sec^2θ =? 7 + tan^2θ + cot^2θ
1 + 4 + cosec^2θ + sec^2θ =? 7 + tan^2θ + cot^2θ
5 + 1 + cot^2θ + 1 + tan^2θ =? 7 + tan^2θ + cot^2θ
7 + cot^2θ + tan^2θ = 7 + tan^2θ + cot^2θ

Answer 4

i'm addressing theta as @ question=sin@(a million+sin@/cos@)+cos@(a million+cos@... =((sin@)^2*cos@+(sin@)^3+(cos@)^2*sin@+(... =(sin@(cos2@+sin2@)+cos@(cos2@+sin2@))/c... =(sin@+cos@)/sin@cos@ =sin@/sin@cos@+cos@/sin@cos@ =sec@+cosec@ sorry for writing sine sq. @ as sin2@ and cos sq. @ as cos2@