Question

Tan square theta + cot squared theta + 2 barabar sec square theta +cos squared theta

Answer 1

Your question needs a correction.

Correct question : tan²θ + cot²θ + 2 = sec²θ + cosec²θ

To Prove : tan²θ + cot²θ + 2 = sec²θ + cosec²θ

     Solution :

⇒ sec²θ + cosec²θ

From trigonometric identities, sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ

⇒ ( 1 + tan²θ ) + ( 1 + cot²θ )

⇒ 1 + tan²θ + 1 + cot²θ

⇒ tan²θ + cot²θ + 2

Hence, proved. Or we can solve it from left hand side too.

⇒ tan²θ + cot²θ + 2

⇒ tan²θ + cot²θ + 1 + 1  

⇒  ( 1 + tan²θ ) + ( 1 + cot²θ )

From trigonometric identities, sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ

⇒  sec²θ + cosec²θ

Hence, proved.

$\:$

Answer 2

Given
tan2A + cot2 A = Sec2A Cosec2A - 2

Taking L.H.S.
⇒tan2A + cot2 A

⇒Sec2A - 1 + Cosec2A - 1 ( As we know 1 + tan2A = Sec2A , 1 + cot2A = Cosec2A )

⇒Sec2A + Cosec2A - 2

⇒1Cos2A + 1Sin2A - 2

⇒Sin2A + Cos2ACos2ASin2A - 2

⇒1Cos2ASin2A - 2

⇒Sec2A Cosec2A - 2
Hence
L.H.S. = R.H.S. ( Hence proved )