Show that tan⁴ θ + tan² θ = sec⁴ θ - sec² θ

Solution:

LHS = tan⁴ θ + tan² θ

Taking out tan² θ as common

= tan² θ (tan² θ + 1)

We know that

1 + tan² θ = sec² θ

i.e. tan² θ = sec² θ - 1

It can be written as

= (sec² θ - 1) sec² θ

So we get

= sec⁴ θ - sec² θ

= RHS

Therefore, it is proved.

 ✦ Try This: Prove that sin² θ/cos θ + cos θ = sec θ

LHS = sin² θ/cos θ + cos θ

By taking LCM

= (sin² θ + cos² θ)/ cos θ

We know that

sin² θ + cos² θ = 1

= 1/cos θ

= sec θ

= RHS

Therefore, it is proved.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.3 Problem 15

Show that tan⁴ θ + tan² θ = sec⁴ θ - sec² θ

Summary:

Trigonometric ratios can be used to determine the ratios of any two sides out of a total of three sides of a right-angled triangle in terms of the respective angles. It is shown that tan⁴ θ + tan² θ = sec⁴ θ - sec² θ

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