Question

Prove that (tan²y) (cos²y) + 1/sec²y = 1​

Answer 1

Explanation:

`(\tan^2y) (\cos^2y) +(1)/(\sec^2y) = 1\\</p><p>LHS </p><p>= (\tan^2y) (\cos^2y) +(1)/(\sec^2y) \\\\</p><p>(\sin^2y)/(\cos^2y)* \cos^2 y + \cos^2 y \\\\</p><p>= \sin^2y + \cos^2 y \\\\</p><p>= 1\\\\</p><p>= RHS\\\\</p><p>\purple {\boxed {\bold {\therefore (\tan^2y) (\cos^2y) +(1)/(\sec^2y) = 1}}} \\`

Hence Proved

Answer 2

see explanation

Explanation:

Using the trigonometric identities

tan x =
`(sinx)/(cosx)` , sec x =
`(1)/(cosx)` , sin²x + cos²x = 1

Consider the left side

tan²y × cos²y +
`(1)/(sec^2y)`

=
`(sin^2y)/(cos^2y)` × cos²y +
`(1)/((1)/(cos^2y) )` ← cancel cos²y in the product

= sin²y + cos²y

= 1

= right side ⇒ proven