2022-04-02 Answered
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Answers (1)

user_27qwe
Answered 2022-04-21 Author has 208 answers

tan2(α)-sin2(α)=tan2(α)sin2(α) Start on the left side.

tan2(α)-sin2(α)

Convert to sines and cosines.

Write tan(α) in sines and cosines using the quotient identity.

(sin(α)cos(α))2-sin2(α)

Apply the product rule to sin(α)cos(α).

sin2(α)cos2(α)-sin2(α)

Write -sin2(α) as a fraction with denominator 1.

sin2(α)cos2(α)+-sin2(α)1

Add fractions.

To write -sin2(α)1 as a fraction with a common denominator, multiply by cos2(α)cos2(α).

sin2(α)cos2(α)+-sin2(α)1cos2(α)cos2(α)

Multiply -sin2(α)1 by cos2(α)cos2(α).

sin2(α)cos2(α)+-sin2(α)cos2(α)cos2(α)

Combine the numerators over the common denominator.

sin2(α)-sin2(α)cos2(α)cos2(α)

Apply Pythagorean identity.

Multiply by 1.

sin2(α)1-sin2(α)cos2(α)cos2(α)

Factor sin2(α) out of -sin2(α)cos2(α).

sin2(α)1+sin2(α)(-cos2(α))cos2(α)

Factor sin2(α) out of sin2(α)1+sin2(α)(-cos2(α)).

sin2(α)(1-cos2(α))cos2(α)

Apply pythagorean identity.

sin2(α)sin2(α)cos2(α)

Multiplysin(α)2 by sin(α)2 by adding the exponents.

sin4(α)cos2(α)

Rewrite sin4(α)cos2(α) as tan2(α)sin2(α).

tan2(α)sin2(α)

Because the two sides have been shown to be equivalent, the equation is an identity.

tan2(α)-sin2(α)=tan2(α)sin2(α) is an identity

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