Question #cdb8e
1 Answer
See the answer below...
Explanation:
#(1/(sec^2a-cos^2a)+1/(csc^2a-sin^2a))cos^2asin^2a#
#=(1/(1/cos^2a-cos^2a)+1/((1/sin^2a-sin^2a)))cos^2asin^2a#
#=(cos^2a/(1-cos^4a)+sin^2a/(1-sin^4a))cdot cos^2 cdot sin^2a# [in first fraction, multiply both numerator and denominator by#cos^2a# and in second fraction multiply both numerator and denominator by#sin^2a# ]
#={cos^2a/((1-cos^2a)(1+cos^2a))+sin^2a/((1-sin^2a)(1+sin^2a))}cdot cos^2 cdot sin^2a#
#={cos^2a/(sin^2acdot(1+cos^2a))+sin^2a/(cos^2a cdot(1+sin^2a))}cdot cos^2 cdot sin^2a#
[As#color(red)(sin^2theta+cos^2theta=1# ]
#={cos^4a/((1+cos^2a))+sin^4a/((1+sin^2a))}#
#=(cos^4a(1+sin^2a)+sin^4a(1+cos^2a))/((1+cos^2a)(1+sin^2a)#
#=(cos^4a+cos^4a cdot sin^2a+sin^4a+sin^4a cdot cos^2a)/(1+sin^2a+cos^2a+sin^2a cdot cos^2a)#
#=((sin^4a+cos^4a)+sin^2a cdot cos^2a(sin^2a+cos^2a))/(2+sin^2a cdot cos^2a#
#=({(sin^2a)^2+(cos^2a)^2}+sin^2a cdot cos^2a cdot 1)/(2+sin^2a cdot cos^2a#
#=({(sin^2a+cos^2a)^2-2 cdot sin^2a cdot cos^2a}+sin^2a cdot cos^2a cdot 1)/(2+sin^2a cdot cos^2a#
#=(1-sin^2a cdot cos^2a)/(2+sin^2a cdot cos^2a#
