Question

Tana + cot a = sec a . cosec a

Answer 1

Step-by-step explanation:

To Prove : tan A + cot A = sec A . cosec A

Proof : L.H.S. = tan A + cot A

${\boxed{\tt{\red{tan \ A = {\dfrac{sin \ A}{cos \ A}} }}}}$

${\boxed{\tt{\red{cot \ A = {\dfrac{cos \ A}{sin \ A}} }}}}$

→ ${\sf{ {\dfrac{sin \ A}{cos \ A}} + {\dfrac{cos \ A}{sin \ A}}}}$

→ ${\sf{ {\dfrac{sin^2 \ A + cos^2 \ A}{(cos \ A)(sin \ A)}}}}$

${\boxed{\tt{\red{sin^2 \ A + cos^2 \ A = 1}}}}$

→ ${\sf{ {\dfrac{1}{(cos \ A)(sin \ A)}}}}$

→ ${\sf{ {\dfrac{1}{cos \ A}} . {\dfrac{1}{sin \ A}} }}$

${\boxed{\tt{\red{ {\dfrac{1}{cos \ A}} = sec \ A}}}}$

${\boxed{\tt{\red{ {\dfrac{1}{sin \ A}} = cosec \ A}}}}$

→ sec A . cosec A

= R.H.S.

Hence, proved !!

Other trigonometric identities which are frequently used :

• sin A = Perpendicular / Hypotenuse
• cos A = Base / Hypotenuse
• tan A = Perpendicular / Base
• sec A = Hypotenuse / Base
• cosec A = Hypotenuse / Perpendicular
• cot A = Base / Perpendicular
• tan²A + 1 = sec²A
• cot²A + 1 = cosec²A