Cos^2 a( cos^4 a+ cos^2 a. Sin ^2a + sin^2 a+ tan ^2 a) = 1

To prove that cos^2 a( cos^4 a+ cos^2 a. Sin ^2a + sin^2 a+ tan ^2 a) = 1, we can simplify the expression step by step:

cos^2 a( cos^4 a+ cos^2 a. Sin ^2a + sin^2 a+ tan ^2 a)

Using the distributive property, we can expand the expression:

= cos^6 a + cos^4 a * cos^2 a * sin^2 a + cos^2 a * sin^2 a + cos^2 a * tan^2 a

Next, we can use the trigonometric identity cos^2 a + sin^2 a = 1:

= cos^6 a + cos^4 a * cos^2 a * sin^2 a + 1 * cos^2 a + cos^2 a * tan^2 a

Now, we can simplify further by factoring out cos^2 a:

= cos^2 a (cos^4 a + cos^2 a * sin^2 a + 1 + tan^2 a)

Using the trigonometric identity 1 + tan^2 a = sec^2 a, we can substitute it in:

= cos^2 a (cos^4 a + cos^2 a * sin^2 a + sec^2 a)

Next, we can use the Pythagorean identity sin^2 a + cos^2 a = 1:

= cos^2 a (cos^4 a + (1 - cos^2 a) * sin^2 a + sec^2 a)

Simplifying further:

= cos^2 a (cos^4 a + sin^2 a - cos^2 a * sin^2 a + sec^2 a)

Using the trigonometric identity 1 - cos^2 a = sin^2 a, we can substitute it in:

= cos^2 a (cos^4 a + sin^2 a - sin^2 a * sin^2 a + sec^2 a)

Simplifying again:

= cos^2 a (cos^4 a + sec^2 a)

Using the trigonometric identity sec^2 a = 1 + tan^2 a, we can substitute it in:

= cos^2 a (cos^4 a + 1 + tan^2 a)

Finally, using the trigonometric identity cos^2 a + sin^2 a = 1, we can substitute it in:

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

= cos^2 a * sec^2 a

Using the trigonometric identity sec^2 a = 1/cos^2 a, we can substitute it in:

= cos^2 a * (1/cos^2 a)

= 1

Therefore, cos^2 a( cos^4 a+ cos^2 a. Sin ^2a + sin^2 a+ tan ^2 a) = 1.