Question

cos to the power 4 alpha by cos square theta + sin to the power 4 theta is equal to 1 then prove that sin to the power 4 alpha + sin to the power 4 theta is equal to 2 sin square theta into sin square theta​

Answer 1

Answer:

Step-by-step explanation: Cos⁴α /Cos²θ  + Sin⁴α/Sin²θ  = 1

to be proved

Cos⁴θ /Cos²α  + Sin⁴θ/Sin²α  = 1

Cos⁴α /Cos²θ  + Sin⁴α/Sin²θ  = 1

=> Cos⁴αSin²θ  + Sin⁴αCos²θ = Cos²θSin²θ

=> Cos⁴α(1 - Cos²θ)  + (1 - Cos²α)²Cos²θ = Cos²θ(1 - Cos²θ)

=>  Cos⁴α - Cos⁴αCos²θ  + (1  + Cos⁴α - 2Cos²α)Cos²θ = Cos²θ - Cos⁴θ

=>Cos⁴α - Cos⁴αCos²θ  + Cos²θ  + Cos⁴αCos²θ - 2Cos²αCos²θ = Cos²θ - Cos⁴θ

=> Cos⁴α  - 2Cos²αCos²θ + Cos⁴θ  = 0

=> (Cos²α - Cos²θ)² = 0

=> Cos²α - Cos²θ = 0

=>  Cos²α = Cos²θ

LHS

Cos⁴θ /Cos²α  + Sin⁴θ/Sin²α

= (Cos²α)²/Cos²α  + (1 - Cos²θ)²/(1 -Cos²α)

= Cos²α + (1 - Cos²α)²/(1 -Cos²α)

=  Cos²α + 1 - Cos²α

= 1

= RHS

QED

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Answer 2

Answer:

Step-by-step explanation:

Step-by-step explanation: Cos⁴α /Cos²θ  + Sin⁴α/Sin²θ  = 1

to be proved

Cos⁴θ /Cos²α  + Sin⁴θ/Sin²α  = 1

Cos⁴α /Cos²θ  + Sin⁴α/Sin²θ  = 1

=> Cos⁴αSin²θ  + Sin⁴αCos²θ = Cos²θSin²θ

=> Cos⁴α(1 - Cos²θ)  + (1 - Cos²α)²Cos²θ = Cos²θ(1 - Cos²θ)

=>  Cos⁴α - Cos⁴αCos²θ  + (1  + Cos⁴α - 2Cos²α)Cos²θ = Cos²θ - Cos⁴θ

=>Cos⁴α - Cos⁴αCos²θ  + Cos²θ  + Cos⁴αCos²θ - 2Cos²αCos²θ = Cos²θ - Cos⁴θ

=> Cos⁴α  - 2Cos²αCos²θ + Cos⁴θ  = 0

=> (Cos²α - Cos²θ)² = 0

=> Cos²α - Cos²θ = 0

=>  Cos²α = Cos²θ

LHS

Cos⁴θ /Cos²α  + Sin⁴θ/Sin²α

= (Cos²α)²/Cos²α  + (1 - Cos²θ)²/(1 -Cos²α)

= Cos²α + (1 - Cos²α)²/(1 -Cos²α)

=  Cos²α + 1 - Cos²α

= 1

= RHS

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