# sin theta sec^7 theta+cos theta cosec^7 theta#
# =sintheta/costheta sec^6 theta+costheta/sintheta cosec^6 theta#
# =tantheta (1+tan^2theta)^3 +1/tantheta (1+cot^2theta)^3#
# =tantheta (1+tan^2theta)^3 +1/tantheta (1+1/tan^2theta)^3#
# =sqrt(a/b) (1+a/b)^3 +sqrt(b/a) (1+b/a)^3#
# =sqrt(a/b)xx (a+b)^3/b^3 +sqrt(b/a) xx(a+b)^3/a^3#
# =(a+b)^3(sqrta/b^(7/2) +sqrtb/a^(7/2))#
# =((a+b)^3(a^4+b^4))/((ab)^(7/2))#
Q-2
Let
#tanA=3 and tanB=2#
So #A>B#
We are to find out
#sin2(A-B)#
Expanding we get
#sin2(A-B)=sin2Acos2B-cos2Asin2B#
So we are to know
#sin2A,cos2B,cos2A and sin2B#
Now #sin2A=(2tanA)/(1+tan^2A)#
#=>sin2A=(2xx3)/(1+3^2)=3/5#
And
#sin2B=(2tanB)/(1+tan^2B)#
#=>sin2B=(2xx2)/(1+2^2)=4/5#
#cos2A=(1-tan^2A)/(1+tan^2A)#
#=>cos2A=(1-3^2)/(1+3^2)=-4/5#
And
#cos2B=(1-tan^2B)/(1+tan^2B)#
#=>cos2B=(1-2^2)/(1+2^2)=-3/5#
Now inserting the values we get
#sin2(A-B)#
#=sin2Acos2B-cos2Asin2B#
#=3/5xx(-3/5)-(-4/5)xx4/5#
#=-9/25+16/25=7/25#
