Question

If cos theta =3/5, find the value of 5 cosec theta - 4 tan theta /​ sec theta + cot theta​

Answer 1

COS THETA=3/5 THEN SEC THETA=5/3 .WKT SIN²THETA+COS²THETA=1      NOW sin²theta=1-cos²theta⇒sintheta=√1-cos²theta=√1-(3/5)²=√16/25=4/5    now cosectheta=5/4, now tantheta= sintheta/costheta=4/5÷3/5=4/3                         ⇒5(5/4)-4(4/3)÷5/3+3/4=(25/4-16/3)÷(5/3+3/4)=75-64/12÷25+12/12                      =11/37

Answer 2

Given, Cosθ = 3/5
According to Pythagoras theorem, in a right-angled triangle, Cosθ = adjacent side/hypotenuse
i.e, adj.side = 3 and hypotenuse = 5
But, (hypotenuse)^2 = (adj.side)^2+(opp.side)^2
=> (opp.side)^2 = (hypotenuse)^2 - (adj.side)^2 = (5)^2 - (3)^2 = 25-9 = 16
therefore, opp.side = 4
And, Sinθ = opp.side/hyp = 4/5
Cosθ = adj.side/hyp = 3/5
Tanθ = opp.side/adj.side = 4/3
Cotθ = adj.side/opp.side = 3/4
Secθ = hyp/adj.side = 5/3
Cosecθ = hyp/opp.side = 5/4
Now, (5Cosecθ-4Tanθ)/(Secθ+Cotθ) = (5(5/4)-4(4/3))/(5/3+3/4)
                                                           = (75-64)/(20+9) = (11/29)