Prove that secq(1-sinq)(secq + tanq) = 1

Here we have to prove secq(1-sinq)(secq + tanq) = 1 we will start with LHS (Left hand side ie secq(1-sinq)(secq + tanq) ) and prove that it is equal to 1 Hence we will get LHS = RHS
Our aim is to Show that secq(1-sinq)(secq + tanq) = 1

Proof

Lets start with LHS of the equations
LHS = secq(1-sinq) (secq + tanq)

= (secq-sinq×secq) (secq+tanq)

= (secq-tanq) (secq+tanq) hint : secq=1/cosq and sinq/cosq=tanq

= (sec²q-tan²q) hint : (x+y)(x-y)=x²-y²

= 1 hint : we know sec²q-tan²q =1
= RHS
Hence proved RHS = LHS ie secq(1-sinq)(secq + tanq) = 1

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

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

Prove that tanq+Secq-1÷tanq-secq+1=cosq÷1-sinq

Here am going to explain the steps Prove that tanq+Secq-1÷tanq-secq+1=cosq÷1-sinq

LHS => tanq+Secq-1/tanq-secq+1

=> (sinQ/cosQ+1/cosQ-1)/(sinQ/cosQ-1/cosQ+1) (divide nominator and denominate by cosQ)

=> (sinQ+1-cosQ)/(sinQ-1+cosQ) (Multipy nominator and denominate by (1-sinQ))

=> (1-sinQ)(sinQ+1-cosQ)/(1-sinQ)(sinQ-1+cosQ)

=> (sinQ+1-cosQ-sinQ2-sinQ+sinQ*cosQ)/(1-sinQ)(sinQ-1+cosQ)

=> (1-cosQ-sinQ2+sinQ*cosQ)/(1-sinQ)(sinQ-1+cosQ)
=> (1-sinQ2-cosQ+sinQ*cosQ)/(1-sinQ)(sinQ-1+cosQ)

=> (cosQ2-cosQ+sinQ*cosQ)/(1-sinQ)(sinQ-1+cosQ)
=> cosQ(cosQ-1+sinQ)/(1-sinQ)(sinQ-1+cosQ)

=> cosQ(cosQ-1+sinQ)/(1-sinQ)(cosQ-1+sinQ)
=> cosQ/(1-sinQ) = RHS

Hence proved