Prove the identity: 1+tan^2(x)= (1/cos^2(x))?

Answer 1

1 + tan^2(x) = sec^2(x) = 1/cos^2(x).

Answer 2

Use the identity 1+tan^2(x)=sec^2(x)
this implies that tan^2(x)=sec^2(x)-1

substituting...

1+sec^2(x)-1=1/cos^2(x)

1 will cancel

sec^2(x)=1/cos^2(x)

Reciprocal identitites: sec x = 1/cos x

so...

1/cos^2(x) = 1/cos^2(x)■

Answer 3

1 + tan^2(x) = sec^2(x) = 1/cos^2(x)

Answer 4

1 + tan ² x
= 1 + sin ² x / cos ² x
= (cos ² x + sin ² x) / cos² x
= 1 / cos ² x

Answer 5

LHS=1+tan^2(x)
=1+[sin^2(x)/cos^2(x)] because [ tan(x)=sin(x)/cos(X)]
=[cos^2(x)+sin^2(x)]/cos^2(x) taking the common denominator

=1/cos^2(x) as cos^2(x)+sin^2(x)=1

Answer 6

since we know that 1+tan^2(x)=sec^2(x), there for:

sec^2(x)=(1/cos^2(x))

we know that sec^2(x)=1/cos^2(x)

there for we prove that 1/cos^2(x)=1/cos^2(x).

Answer 7

(1/cos(x))^2=(sec(x))^2
=1+(tan(x))^2 as required
{pythagoras}

i hope that this helps

Answer 8

well its simple

we all know
sin^2(x)+ cos^2(x)=1

now divide both sides by cos^2(x)
we get

tan^2(x)+1= 1/cos^2(x)

Answer 9

1+tan^2(x)
=1+sin^2(x)/cos^2(x)
={cos^2(x)+sin^2(x)}/cos^2(x)

Now, cos^2(x)+sin^2(x)=1

Therefore,
{cos^2(x)+sin^2(x)}/cos^2(x)
=1/cos^2(x)

So,
1+tan^2(x)=1/cos^2(x)

Answer 10

multiply both sides by cos^2(x)

cos^2(x) + tan^2(x)cos^2(x)=1

cos^2(x) + (sin^2(x)cos^2(x))/cos^2(x)=1

cos^2(x) + sin^2(x)=1

cos^2(x) + (1-cos^2(x))=1

1=1
IS TRUE