Negative Angle Identities

sin (–x) = – sin x

cos (–x) = cos x

Note: Sometimes, these identities are called opposite angle identities

Since

  cos (–x) = cos x

i.e. value of cos remains same after –x,

it is called even function

Since

  sin (–x) = – sin x

i.e. value of sin becomes negative after –x,

 it is called odd function

Thus, it is called odd-even identities also

We need to remember only these two,

for tan, cot, cosec, sec we can find using these two.

Let’s see

tan

Now, since tan = sin/cos

tan (–x) = sin (–x)/cos (–x)

       = – sin x/cos x

       = – tan x

∴ tan (–x) = – tan x

sec

Now, since sec = 1/cos

sec (–x) = 1/cos (–x)

  = 1/cos x

  = sec x

∴ sec (–x) = sec x

cosec

Now, since cosec = 1/sin

cosec (–x) = 1/sin (–x)

       = 1/(– sin x)

       = – 1/sin x

       = – cosec x 

∴ cosec (–x) = – cosec x

cot

Now, since cot = 1/tan

cot (–x) = 1/tan (–x)

  = 1/(– tan x)

  = – 1/tan x

  = – cot x

∴ cot (–x) = – cot x