Question

Prove that
(1-tan A)2 + (1 - cot A)2 = (sec A- cosec A).​

Answer 1

Step-by-step explanation:

(1-tanA)²+(1-cotA)²

=(cosA-sinA/cosA)²+(sinA-CosA/sinA)²

=(cosA-sinA/CosA)²+ {-(cosA-sinA/sinA)}²

=(cosA-sinA/cosA)²+(cosA-sinA/sinA)²

=(cosA-sinA)²sin²A+(cosA-sinA)²cos²A/sin²A cos²A

=(cosA-sinA)²(sin²A+cos²A)/sin²A cos²A

=(cosA-sinA)²/sin²A cos²A

=(1/sinA-1/cosA)²

=(cosecA-secA)²

=(secA-cosecA)²

Answer 2

Step-by-step explanation:

How do you prove that sinA(1+tanA)+cosA(1+cotA)=secA+cosecA?

To prove sin A(1+ tan A)+ cos A(1 + cot A) = sec A + cosec A.

LHS = sin A(1+ tan A)+ cos A(1 + cot A)

= sin A + sin^2 A/ cos A + cos A + cos^2 A/ sin A

= sin A + cos A + [sin^3 A + cos^3 A]/sin A cos A

=[ sin^2 A cos A + cos^2 A sin A + sin^3 A + cos^3 A]/sin A cos A

= [ sin^2 A cos A +cos^3 A + cos^2 A sin A + sin^3 A]/sin A cos A

= [cos A (sin^2 A + cos^2 A) + sin A (sin^2 A + cos^2 A)]/sin A cos A

= [cos A +sin A]/sin A cos A

= (1/sin A) + (1/cos A)

= cosec A + sec A = RHS.

Proved.