prove trigonometric identities

Question

secxtanx=cscxcotx

Answer 1

Sec x tan x = csc x cot x

Left hand side identity = Sec x tan x

=  1 / cos x * ( sin x / cos x ) 

= sin x / cos ² x

Left hand side identity = sin x / cos ² x

Right hand side identity = csc x cot x

=  1 / sin x * (cos x / sin x ) 

= cos x / sin ² x

Right hand side identity = cos x / sin ² x

Left hand side identity  = Right hand side identity

sin x /cos ² x  =  cos x / sin ² x

sin ³ x = cos ³ x 

Let  us assume that  x = π / 4

sin ³ x = sin ³ ( π / 4 ) =  ( 1/ √ 2)³ = 0 . 3535 

cos ³ x = cos ³ (π / 4) = ( 1/ √ 2)³ = 0 . 3535

sin ³ x = cos ³ x

Since period of a sinusoidal function is 2π.

So. it is also true for all x = π / 4 + 2nπ where n is integer.

For all  x values except  x = π / 4 + 2nπ  the identity is not true.

Therefore Sec x tan x ≠ csc x cot x .