Question

show that (sin A + cos ecA)^2 + (cos A+ sec A)^2 = 7+ tan^2 A+ cot^2 A​

Answer 1

Step-by-step explanation:

(sinA+cosecA) ²+(cosA+secA) ²=7+tan ²A+cot ²A.

(sinA+cscA) ²+(cosA+secA) ²

=sin ²A+csc ²A+2sinAcscA+cos ² A+sec²A+2cosAsecA

.......As[a²+b²+2ab=(a+b)²]

=sin ²A+csc ² A+2sinA× 1/sinA +cos ²A+sec ² A+2cosA 1/cosA

........... since secA= 1/cosA

and cscA= 1/sin A

=sin² A+csc ²A+2+cos ²A+sec ²A+2

=(sin ² A+cos ² A)+csc ² A+sec ²A+4

=1+1+cot 2A+1+tan 2A+4 ........... since csc 2A=1+cot 2 A and sec ²A=1+tan ² A

=7+tan ²A+cot ²A

Hence proved.