Question

If cos θ=12/13, show that sinθ(1−tan θ)=35/156

Answer 1

Given
cosθ=12/13 …. (1)

To show that sinθ(1–tanθ)=35/156

Now we know that cosθ=Base/ Hypotenuse ….(2)

Therefore, by comparing equation (1) and (2)

We get,

Base side adjacent to ∠θ = 12

And

Hypotenuse = 13

Therefore from above figure

Base side BC= 12

Hypotenuse AC= 13

Side AB is unknown and it can be determined by using Pythagoras theorem

Therefore by applying Pythagoras theorem

We get,

AC²= AB² + BC²

Therefore by substituting the values of known sides

We get,

13²= AB² + 12²

Therefore,

AB²= 13²– 12²

AB²= 169 – 144

AB =25

AB= √25

Therefore,

AB = 5 …. (3)

Now, we know that

sinθ=Perpendicular side opposite to ∠θ /Hypotenuse

Now from figure (a)

We get,

sinθ=AB/AC

Therefore,

sinθ = 5/13 …. (5)

Now L.H.S of the equation to be proved is as follows

L.H.S = sinθ(1–tanθ] …. (6)

Substituting the value of sinθ and tanθ from equation (4) and (5)

We get,

L.H.S = 5/13(1−5/12)

Taking L.C.M inside the bracket

We get,

L.H.S= 5/13((1×12−5)/12)

Therefore,

L.H.S= (5/13)((12–5)/12)

L.H.S = (5/13)(7/12)

Now by opening the bracket and simplifying

We get,

L.H.S = (5×7)/(13×12)

L.H.S= 35/136

From equation (6) and (7) ,it can be shown that

that sinθ(1–tanθ) = 35/136