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Writing trigonometric compositions algebraically: Prove each of the following equalities, which rewrite compositions of trigonometric and inverse trigonometric functions as algebraic functions.

cossin-1x=1-x2sincos-1x=1-x2sec2tan-1x=1+x2tansec-1x=x1-1x2

Short Answer

Expert verified

The four equalities:

cossin-1x=1-x2sincos-1x=1-x2sec2tan-1x=1+x2tansec-1x=x1-1x2

have been proved.

Step by step solution

01

Step 1. Given Information 

We have been given four equalities and we need to prove them.

cossin-1x=1-x2sincos-1x=1-x2sec2tan-1x=1+x2tansec-1x=x1-1x2

02

Step 2. Proving the first equality

It is known that sin2y+cos2y=1and it can be written as:

cos2y=1-sin2y.

Now we substitute sin-1xfor y.

role="math" localid="1654264701384" cos2sin-1x=1-sin2sin-1xcossin-1x2=1-sinsin-1x2cossin-1x2=1-x2cossin-1x=±1-x2

For -π2yπ2, cosyis always positive. So cossin-1x=1-x2.

Thus the first equality is proved.

03

Step 3. Proving the second equality

It is known that sin2y+cos2y=1and it can be written as sin2y=1-cos2y.

Now we substitute cos-1xfor y.

sin2cos-1x=1-cos2cos-1xsincos-1x2=1-coscos-1x2sincos-1x2=1-x2sincos-1x=±1-x2

For 0yπ, sinyis always positive. So sincos-1x=1-x2.

Thus the second equality is proved.

04

Step 4. Proving the third equality

It is known that sec2y-tan2y=1and it can be written as sec2y=1+tan2y.

Now substitute tan-1xfor y

sec2tan-1x=1+tan2tan-1xsec2tan-1x=1+tantan-1x2sec2tan-1x=1+x2

Thus the third equality is proved.

05

Step 5. Proving the fourth equality

It is known that sec2y-tan2y=1and it can be written as role="math" localid="1654265261202" tan2y=sec2y-1.

Now substitute role="math" localid="1654265269848" sec-1xfor y

role="math" localid="1654265385702" tan2sec-1x=sec2sec-1x-1tansec-1x2=secsec-1x2-1tansec-1x2=x2-1tansec-1x=x2-1tansec-1x=x21-1x2tansec-1x=x1-1x2

And so the fourth equality is proved.

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