__maths__>__Trigonometry__>__Trigonometric Identities and Complementary Angles__### Pythagorean Trigonometric Identities

The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"

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For the triangle given in figure, `sin theta = (PQ)/(OQ)`

For the triangle given in figure, `cos theta = (OP)/(OQ)`

The Pythagoras theorem is given as

`OP^2 + PQ^2 =OQ^2`

Or equivalently

`((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1`

For a right angled triangle, the Pythagoras theorem is given as

`(text(opposite))^2` `+ (text(adjacent))^2` `= (text(hypotenuse))^2`

`OP^2 + PQ^2 =OQ^2`

If this equation is divided by `(text(hypotenuse))^2` or `OQ^2`

`((OP)/(OQ))^2 + ((PQ)/(OQ))^2 = 1`

Or equivalently

`sin^2 theta + cos^2 theta = 1`

Pythagorean Theorem states that in a right angled triangle, square of hypotenuse equals sum of squares of two arms. The trigonometric ratios are defined for right angled triangles. The relationships between trigonometric ratios per Pythagorean theorem are called "Pythagorean Trigonometric Identities".

`sin^2 theta + cos^2 theta = 1`

It is noted that the result is true for any value of `theta`. That is, if `theta = 27`, then `sin^2 27^@ + cos^2 27^@ = 1`

The word "identity" means equality of two expressions; left and right hand side are identical.

In the Pythagorean Trigonometric Identity `sin^2 theta + cos^2 theta = 1`, it is stated that left hand side `sin^2 theta + cos^2 theta` equals the right hand side `1`.

For a right angled triangle,

`sin^2 theta + cos^2 theta = 1`

If this equation is divided by `sin^2 theta`, the following identity is derived

`1+cot^2 theta = csc^2 theta`

`1+cot^2 theta = csc^2 theta`

For a right angled triangle,

`sin^2 theta + cos^2 theta = 1`

If this equation is divided by `cos^2 theta`, the following identity is derived.

`tan^2 theta + 1 = sec^2 theta`

`tan^2 theta + 1 = sec^2 theta`

**Pythagorean Trigonometric Identities: **

For any theta,

`sin^2 theta + cos^2 theta = 1`

`1+cot^2 theta = csc^2 theta`

`tan^2 theta + 1 = sec^2 theta`

Note that this need not me memorized, connect these to the Pythogoras theorem and quickly derive when required.

*Solved Exercise Problem: *

What is the value of `1-sin^2 theta`?

- `cos^2 theta`
- `cos^2 theta`
- the question is incomplete to find an answer

The answer is '`cos^2 theta`'.

This is derived from the Pythagorean trigonometric identity `sin^2 theta + cos^2 theta = 1`

*Solved Exercise Problem: *

What is the value of `sec^2 theta - 1`?

- `tan^2 theta`
- `tan^2 theta`
- `cot^2 theta`

The answer is '`tan^2 theta`'.

This is derived from the Pythagorean trigonometric identity `tan^2 theta + 1 = sec^2 theta`

*Solved Exercise Problem: *

What is the value of `csc^2 theta-cot^2 theta`?

- `cos^2 theta`
- 1
- 1

The answer is '`1`'.

This is derived from the Pythagorean trigonometric identity `1 + cot^2 theta = csc^2 theta`

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