# CLASS-9TRIGONOMETRICAL RATIO - SQUARE RELATIONS IN T-RATIOS

Square Relations in T-Ratios -

Theorem-2. In a right angled triangle, for any acute angle θ, we have :-

(i) sin² θ + cos² θ = 1,

(ii) 1 + tan² θ = sec² θ,

(iii) 1 + cot² θ =  cosec² θ

Proof : - Consider a right angled ABC in which ∠B = 90⁰ and ∠A = θ⁰

Let, AB = x units, BC = y units, AC = r units

Then, by the Pythagoras theorem, we have - x² + y² = r²

Y           X

(i)  sin² θ + cos² θ = (-----)² + (-----)²

r            r

y²         x²

=  (------ + ------)

r²         r²

(y² + x²)          r²

=  ------------ = -------  [x² + y² = r²]   = 1

r²              r²

So, sin² θ + cos² θ = 1      (Proved)

y

(ii)  1 + tan² θ =  1 + (-----)²

x

y²         x² + y²

=  1 + ------ = -----------   [x² + y² = r²]

x²           x²

r²         r

= ------ = (-----)²  = sec² θ

x²        x

1 + tan² θ = sec² θ    (Proven)

x

(iii)   1  + cot² θ =  1 + (------)²

y

x²           y² + x²

=  1 +  -------  =  ----------  [x² + y² = r²]

y²              y²

r²               r

=  --------- =  (--------)² =  cosec² θ  (Proved)

y²              y