Prove that (Sinq + Cosecq)2 + (Cosq + Secq)2 = 7 + Tan2q + Cot2q

we have to prove (Sinq + Cosecq)2 + (Cosq + Secq)2 = 7 + Tan2q + Cot2q we will start with LHS (Left hand side) and prove that it is equal to 7 + Tan2q + Cot2q Hence we will get LHS = RHS

Proof

Lets start with LHS of the equations

LHS = (sinq + cosecq)² + (cosq + secq)²
= (sin²q + cosec²q + 2 sinq cosecq ) + ( cos²q + sec²q + 2 cosq secq )
= sin²q + cosec²q + cos²q + sec²q + 2 + 2 Hint :2 * sinq * 1/sinq + 2 * cosq * 1/cosq

= ( sin²q + cos²q) + cosec²q + sec²q + 2 + 2

= 1 +cosec²q + sec²q + 4

= (1 + cot²q) + (1 + tan²q) + 5

= 7 + tan²q + cot²q
= RHS
Hence proved RHS = LHS ie (Sinq + Cosecq)2 + (Cosq + Secq)2 = 7 + Tan2q + Cot2q

See in Picture