Fill in the blank to complete the fundamental identity. [tex]$\tan ^2 \theta+1=$[/tex]
Answer (2)
In trigonometry, one of the important identities is based on the tangent function. The identity is as follows:
tan 2 θ + 1 = sec 2 θ
This identity is derived from the Pythagorean identity:
sin 2 θ + cos 2 θ = 1
To understand how we get tan 2 θ + 1 = sec 2 θ , let's break it down step by step:
Start with the Pythagorean identity: sin 2 θ + cos 2 θ = 1
Divide each term by cos 2 θ : cos 2 θ sin 2 θ + cos 2 θ cos 2 θ = cos 2 θ 1
Simplify:
c o s 2 θ s i n 2 θ = tan 2 θ
c o s 2 θ c o s 2 θ = 1
c o s 2 θ 1 = sec 2 θ
This results in: tan 2 θ + 1 = sec 2 θ
This identity is a fundamental tool in solving many trigonometric problems where tangent and secant functions are involved. It helps simplify expressions and solve equations efficiently.
So, to complete the fundamental identity given in the question, fill in the blank with sec 2 θ .
The fundamental identity in trigonometry is tan 2 θ + 1 = sec 2 θ . This identity can be derived from the Pythagorean identity sin 2 θ + cos 2 θ = 1. Understanding this relationship is crucial for solving many trigonometric problems.
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