Question

Cot(90-theta)÷tan theta+cosec(90-theta)Ãsec theta÷ tan(90-theta)=?

Answer 1

What is the relation among all the trigonometrical ratios of (90° - θ)?

In trigonometrical ratios of angles (90° - θ) we will find the relation between all six trigonometrical ratios.

Let a rotating line OA rotates about O in the anti-clockwise direction, from initial position to ending position makes an angle ∠XOA = θ. Now a point C is taken on OA and draw CD perpendicular to OX or OX'.

Again another rotating line OB rotates about O in the anti-clockwise direction, from initial position to ending position (OX) makes an angle ∠XOY = 90°; this rotating line now rotates in the clockwise direction, starting from the position (OY) makes an angle ∠YOB = θ.

Now, we can observe that ∠XOB = 90° - θ.



Again a point E is taken on OB such that OC = OE and draw EF perpendicular to

OX or OX'.

Since, ∠YOB = ∠XOA

Therefore, ∠OEF = ∠COD.

Now, from the right-angled ∆EOF and right-angled ∆COD we get, ∠OEF = ∠COD and OE = OC.

Hence, ∆EOF ≅ ∆COD (congruent).

Therefore, FE = OD, OF = DC and OE = OC.

Trigonometrical Ratios of (90° - θ)
0Save






In this diagram FE and OD both are positive. Similarly, OF and DC are both positive.

Trigonometrical Ratios of (90° - θ)
0Save






In this diagram FE and OD both are negative. Similarly, OF and DC are both negative.

Trigonometrical Ratios of (90° - θ)
0Save






In this diagram FE and OD both are negative. Similarly, OF and DC are both negative.

Trigonometrical Ratios of (90° - θ)
0Save








In this diagram FE and OD both are positive. Similarly, OF and DC are both negative.

According to the definition of trigonometric ratio we get,

sin (90° - θ) =
FE
OE


sin (90° - θ) =
OD
OC

, [FE = OD and OE = OC, since ∆EOF ≅ ∆COD]

sin (90° - θ) = cos θ