Verify Trigonometric Identities

How to verify trigonometric identities.

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 θ. There are some fundamental trigonometric identities which are used to prove further complex identities.

We can use any of the given basic identities or formulas to prove any identity or evaluating value of any trigonometric function.

Lets look at some examples:

Example1: prove the given identity.

$\dpi{120} {\color{Red} sin^{3}x + sinx cos^{2}x =sinx}$

Solution: Solving left side we get

$\dpi{120} sinx(sin^{2}x+cos^{2}x)=sinx$ [ factor out sinx]

sinx (1) = sinx [used Pythagorean identity]

sinx = sinx $\dpi{120} \left [sin^{2}x+cos^{2}x=1 \right ]$

Example2: Verify the given identity.

$\dpi{120} {\color{Red} sin^{2}x-sin^{4}x=cos^{2}x-cos^{4}x}$

Solution: Starting with left side,

$\dpi{120} sin^{2}x-sin^{4}x=sin^{2}x\left ( 1-sin^{2}x \right )$ factored out

$\dpi{120} =sin^{2}x\left \left [cos^{2}x \right \right ]$ Applied Pythagorean identity

$\dpi{120} =\left ( 1-cos^{2}x \right )cos^{2}x$ Applied Pythagorean identity

$\dpi{120} =cos^{2}x-cos^{4}x$ multiply

$\dpi{120} sin^{2}x-sin^{4}x=cos^{2}x-cos^{4}x$

Hence verified the identity.

Example3: Verify the given identity

$\dpi{120} {\color{Red} \frac{(1+sinx)}{cosx}+\frac{cosx}{1+sinx}=2secx}$

Solution: Simplifying left side,

$\dpi{120} \frac{(1+sinx)}{cosx}+\frac{cosx}{1+sinx}=\frac{(1+sinx)}{cosx}*\frac{1+sinx}{1+sinx}+\frac{cosx}{1+sinx}*\frac{cosx}{cosx}$ [made the denominators same]

$\dpi{120} =\frac{(1+sinx)^{2}}{cosx(1+sinx)}+\frac{cos^{2}x}{cosx(1+sinx)}$

$\dpi{120} =\frac{(1+sinx)^{2}+cos^{2}x)}{cosx(1+sinx)}$

$\dpi{120} =\frac{1+sin^{2}x+2sinx+cos^{2}x}{cosx(1+sinx)}$

$\dpi{120} =\frac{1+sin^{2}x+cos^{2}x+2sinx}{cosx(1+sinx)}$ [rearranged the terms]

$\dpi{120} =\frac{2+2sinx}{cosx(1+sinx)}$

$\dpi{120} =\frac{2(1+sinx)}{cosx(1+sinx)}$ [ factored out 2]

= 2/cosx = 2secx

Hence verified the identity.

Example4: Verify the following identity.

$\dpi{120} {\color{Red} tan\left ( sin^{-1}\left ( \frac{x-1}{4} \right ) \right )=\frac{x-1}{\sqrt{16-(x-1)^{2}}}}$

Solution: Here we need to use trigonometric substitution.

Let x-1= 4 sinθ and we need to work on both sides .

Working on both sides simultaneously ,

$\dpi{120} tan\left ( sin^{-1}\left ( \frac{4sin\theta }{4} \right ) \right )=\frac{4sin\theta }{\sqrt{16-\left ( 4sin\theta \right )^{2}}}$

$\dpi{120} tan\left ( sin^{-1} sin\theta \right )=\frac{4sin\theta }{\sqrt{16-16sin^{2}\theta }}$

$\dpi{120} tan\theta =\frac{4sin\theta }{\sqrt{16(1-sin^{2}\theta )}}$

$\dpi{120} tan\theta =\frac{4sin\theta }{\sqrt{16cos^{2}\theta }}$

$\dpi{120} tan\theta =\frac{4sin\theta }{4cos\theta }$

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

tanθ = tanθ

Hence verified the identity.

Example5: Use the cofunction identities to evaluate the expression without using a calculator.

$\dpi{120} {\color{Red} cos^{2}20+cos^{2}52+cos^{2}38+cos^{2}70}$

Solution:

We notice that 20 and 70 are complimentary angles, same way 52 and 38 are complimentary angle.

So we can use them for co function identities.

$\dpi{120} cos^{2}20+cos^{2}70+cos^{2}52+cos^{2}38$

$\dpi{120} cos^{2}20+\left (cos70 \right )^{2}+cos^{2}52+\left (cos38 \right )^{2}$

$\dpi{120} cos^{2}20+\left (cos(90-20) \right )^{2}+cos^{2}52+\left (cos(90-52) \right )^{2}$

$\dpi{120} cos^{2}20+\left (sin20 \right )^{2}+cos^{2}52+\left (sin52 \right )^{2}$

$\dpi{120} cos^{2}20+sin^{2}20 +cos^{2}52+sin^{2}52$

$\dpi{120} \left [cos^{2}20+sin^{2}20 \right ] +\left [cos^{2}52+sin^{2}52 \right ]$

1 + 1 = 2

Practice problems:

Verify the following identities:

$\dpi{120} \frac{sec\theta -1}{1-cos\theta }= sec\theta$

(1+siny)[1+sin(-y)] = cos^2(y)

$\dpi{120} tan\left ( cos^{-1}\frac{x+1}{2} \right )=\frac{\sqrt{4-(x+1)^{2})}}{x+1}$

$\dpi{120} tan^{2}63+ cot^{2}16-sec^{2}74-csc^{2}27$ [answer: -2]

How to verify trigonometric identities.

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