All Trigonometric Identities and Formulas

All Trigonometric Identities and Formulas

Trigonometric identities are those equations which are true for all those angles for which functions are defined.

The equation sinθ = cosθ is a trigonometric equation but not a trigonometric identity because it doesn’t hold for all values of There are some fundamental trigonometric identities which are used to prove further complex identities.

Here is a list of all basic identities and formulas.

Pythagorean identities:

$\dpi{120} \mathbf{sin^{2}\theta +cos^{2}\theta =1}$
$\dpi{120} \mathbf{1+tan^{2}\theta =sec^{2}\theta }$
$\dpi{120} \mathbf{1+cot^{2}\theta =csc^{2}\theta}$

Reciprocal identities:

$\dpi{120} \mathbf{sin\theta =\frac{1}{csc\theta } }$ or $\dpi{120} \mathbf{csc\theta =\frac{1}{sin\theta }}$
$\dpi{120} \mathbf{cos\theta =\frac{1}{sec\theta }}$ or $\dpi{120} \mathbf{sec\theta =\frac{1}{cos\theta }}$
$\dpi{120} \mathbf{tan\theta =\frac{1}{cot\theta }}$ or $\dpi{120} \mathbf{cot\theta =\frac{1}{tan\theta }}$

Quotient identities:

$\dpi{120} \mathbf{tan\theta =\frac{sin\theta }{cos\theta }}$ or $\dpi{120} \mathbf{cot\theta =\frac{cos\theta }{sin\theta }}$

Even-Odd identities:

Only cos and sec are even functions ,rest are all odd.

Even: cos(-x) = cos(x) sec(-x) = sec(x)

Odd: sin(-x) =-sin(x) csc(-x) =-csc(x)

tan(-x)=-tan(x) cot(-x) =-cot(x)

Sum and difference formulas:

sin(u±v) = sin(u)cos(v)±cos(u)sin(v)
cos(u±v) = cos(u)cos(v)∓sin(u)sin(v)
$\dpi{120} \mathbf{tan(u\pm v)=\frac{tan(u)\pm tan(v)}{1\mp tan(u)tan(v)}}$

Double angle identities:

sin(2θ)= 2sinθ cosθ

$\dpi{120} \mathbf{cos(2\theta )= cos^{2}\theta -sin^{2}\theta }$

$\dpi{120} \mathbf{cos(2\theta )=2cos^{2}\theta -1}$

$\dpi{120} \mathbf{cos(2\theta )=1-2sin^{2}\theta }$

$\dpi{120} \mathbf{tan(2\theta )= \frac{2tan\theta }{1-tan^{2}\theta }}$

Half angle identities :

$\dpi{120} \mathbf{sin(\frac{\theta }{2})=\pm \sqrt{\frac{1-cos\theta }{2}}}$

$\dpi{120} \mathbf{cos(\frac{\theta }{2})=\pm \sqrt{\frac{1+cos\theta }{2}}}$

$\dpi{120} \mathbf{tan(\frac{\theta }{2})= \pm \sqrt{\frac{1-cos\theta }{1+cos\theta }}=\frac{sin\theta }{1+cos\theta }=\frac{1-cos\theta }{sin\theta }}$

Product to sum identities:

2sin(x)cos(y) = sin(x+y)+sin(x-y)

2cos(x)sin(y) = sin(x+y)-sin(x-y)

2cos(x)cos(y) = cos(x+y)+cos(x-y)

2sin(x)sin(y) = cos(x-y)-cos(x+y)

Sum to product identities:

$\dpi{120} \mathbf{sinx+siny =2sin(\frac{x+y}{2})cos\left ( \frac{x-y}{2} \right )}$

$\dpi{120} \mathbf{sinx-siny =2sin(\frac{x-y}{2})cos\left ( \frac{x+y}{2} \right )}$

$\dpi{120} \mathbf{cosx+cosy=2cos\left ( \frac{x+y}{2} \right )cos\left ( \frac{x-y}{2} \right )}$

$\dpi{120} \mathbf{cosx-cosy=-2sin\left ( \frac{x+y}{2} \right )sin\left ( \frac{x-y}{2} \right )}$ OR

$\dpi{120} \mathbf{cosx-cosy=2sin\left ( \frac{x+y}{2} \right )cos\left ( \frac{y-x}{2} \right )}$

Co-function identities:

$\dpi{120} \mathbf{\mathbf{sin(\frac{\pi }{2}-\theta )=cos\theta }}$ $\dpi{120} \mathbf{\mathbf{cos(\frac{\pi }{2}-\theta )=sin\theta }}$

$\dpi{120} \mathbf{\mathbf{tan(\frac{\pi }{2}-\theta )=cot\theta }}$ $\dpi{120} \mathbf{\mathbf{cot(\frac{\pi }{2}-\theta )=tan\theta }}$

$\dpi{120} \mathbf{\mathbf{sec(\frac{\pi }{2}-\theta )=csc\theta }}$ $\dpi{120} \mathbf{\mathbf{csc(\frac{\pi }{2}-\theta )=sec\theta }}$

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