Integration using trigo identities - sin^2,cos^2 etc formulae

Chapter 7 Class 12 Integrals
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Ex 7.3, 13 Integrate the function cosβ‘γ2π₯ β cosβ‘2πΌ γ/cosβ‘γπ₯ β cosβ‘πΌ γ β«1βγcosβ‘γ2π₯ β cosβ‘2πΌ γ/cosβ‘γπ₯ β cosβ‘πΌ γ " " ππ₯γ =β«1β((2 cos^2β‘γπ₯ β 1γ ) β (2 cos^2β‘γπΌ β 1γ ))/(cosβ‘π₯ β cosβ‘πΌ ) ππ₯ =β«1β(2 cos^2β‘γπ₯ β 1γ β 2 cos^2β‘γπΌ + 1γ)/(cosβ‘π₯ β cosβ‘πΌ ) ππ₯ =β«1β(2 cos^2β‘π₯ β 2 cos^2β‘πΌ + 1 β 1)/(cosβ‘π₯ β cosβ‘πΌ ) ππ₯ =β«1β2(cos^2β‘π₯ β cos^2β‘πΌ )/(cosβ‘π₯ β cosβ‘πΌ ) ππ₯ =2β«1β(cos^2β‘π₯ β cos^2β‘πΌ)/(cosβ‘π₯ β cosβ‘πΌ ) ππ₯ =2β«1β((πππ β‘π₯ β πππ β‘πΌ ) (πππ β‘π₯ + πππ β‘πΌ ))/((πππ β‘π₯ β πππ β‘πΌ ) ) ππ₯ =2β«1β(πππ β‘π₯+πππ β‘πΌ ) ππ₯ =2(β«1βπππ β‘π₯ ππ₯+β«1βπππ β‘πΌ ππ₯) =2(β«1βπππ β‘π₯ ππ₯+πππ β‘πΌ β«1β1. ππ₯) =π(πππβ‘π+πβ‘πππ πΆ)" +" πͺ (πππ β‘2π=2 γπππ γ^2β‘πβ1)